CONSTRUCTION OF BINARY LINEAR CODES

Transcription

1 Undergraduate Research Opportunity Programme in Science CONSTRUCTION OF BINARY LINEAR CODES SOH JOO KIAT KENNETH Supervisor : Dr. Xing Chaoping Department of Mathematics National University of Singapore 1999/2000 1

2 Contents 1. Introduction Motivation Basic Terms and Notation Binary Linear Code Motivation Properties of Binary Linear Codes Generator martix and parity-check matrix Some Construction of Binary Linear Codes Motivation Lengthening a Code Extending a Code Taking a Subcode from a Code Shortening a Code Truncating a Code Concatenation of Codes Combination of Codes The u(u + v)-construction

3 3.10 The (G G)-Construction List of Tables 33 Table 1 : Listing of Optimal Binary Linear Codes Table 2 : Results from Lengthening a Code Table 3 : Results from Extending a Code Table 4 : Results from Subcoding Table 5 : Results from Shortening a Code Table 6 : Results from Truncating a Code Table 7 : Results from Concatenation of Codes Table 8 : Results from Combination of Codes Table 9 : Results from The u(u + v)-construction Table 10 : Results from The (G G)-Construction Conclusion 52 Bibliography 53 3

4 Chapter 1 Introduction 1.1 Motivation Coding Theory is the study of methods for efficient and accurate transfer of information from one party to another. The physical medium through which the information is transmitted is called a channel. Telephone lines and the atmosphere are examples of channels. Encoding is the process of adding redundancy digits to the information digits. The encoding process in general is not difficult. The difficult problem is decoding. Often, undesirable disturbances may result in the distortion of transmitted information, and these disturbances are referred to as noise. Coding theory deals with the problem of detecting and correcting transmission errors caused by noise in the channel. Basically, there are five objectives in the study of coding theory. They are: 1. fast encoding of information. 2. easy transmission of encoded messages. 3. fast decoding of received messages. 4

5 4. correction of errors involved in the channels. 5. maximum transfer of information per unit time. Of the five, the most important is the fourth, but in general, it will not be compatible with the other four, and any solution is necessarily a trade-off among the five objectives. In this chapter, some basic definitions and results in coding theory are stated without proofs. They will be used throughout the next two chapters. Here, we will formally state terms like distance and weight, and illustrate their importance in the understanding of the binary linear code. 1.2 Basic Terms and Notation Definition. The set Z 2 = {0, 1}, which is a finite binary set, is also called a code alphabet. 1. A binary word of length n over Z 2 is a sequence u = u 1 u 2...u n with each u i Z 2 for all i. 2. A binary code C is a set of finite sequences of symbols of Z 2 having the same length n. 3. An element of C is called a codeword in C. 5

6 4. The number of codewords in C, denoted by C, is called the size of C. 5. The length ofacodewordisthenumberof0and1. Definition. The weight of a codeword w, denoted by wt(w), is the number of times the digit 1 occurs. For example, wt(110101) = 4 and wt(00000) = 0. The weight of a code C, denoted by w(c), is the minimum weight of all nonzero codewords in C. For example, let C = {0000, 1100, 0011, 0111, 1011, 1101, 1110, 1111}. Then, w(c) =2. Definition. The Hamming distance, or the distance d(w 1,w 2 ), between any 2 codewords w 1 and w 2, is the number of positions in which the 2 codewords disagree. For example, d(01011, 00111) = 2 and d(10110, 10110) = 0. Proposition 1.1. (Properties of Hamming Distance) Let x, y and z be words of length n over Z 2.Then 1. 0 d(x,y) n, 2. d(x,y)=0 x=y, 3. d(x,y)=d(y,x),and 4. d(x,z) d(x,y)+d(y,z). Definition. Let C be a code with at least two codewords. The minimum 6

7 distance d(c) of Cis the smallest distance between distinct codewords. Symbolically, we write d(c) =min{d(c, d) c, d C, c d}. Since c d implies that d(c, d) 1, the minimum distance of a code must be at least 1. 7

8 Chapter 2 Binary Linear Code 2.1 Motivation Most practical error-correcting codes used today, including the Hamming codes, are examples of linear codes. The main aim of this chapter is to explain what this means, and what consequences of linearity make such codes good. One requirement of a linear code is that its alphabet symbols are the elements of a field. In particular, we shall use the field Z 2, since we are dealing with the construction of binary linear codes. Before actually saying what a linear code is, here are just some of the advantages over arbitrary codes. 1. Evaluation of d(c) ismucheasier. 2. Encoding is fast and requires little storage. 3. It is much easier to determine which errors are correctable/detectable. 4. The probability of correct decoding is much easier to calculate. 5. Very efficient decoding techniques exist for linear codes. 8

9 Before we construct some optimal linear codes, we will study some basic properties of binary linear codes. 2.2 Properties of Binary Linear Codes Definition AcodeC Z n p that is also a subspace of Zn p is called a linear code. If C has length n, dimension kand minimum distance d(c) = d,then Cis an [n,k,d]-code. The numbers n, k, and d are called the parameters of the linear code. Lemma 2.1 If C is a binary code, then d(c, d) =wt(c d) for all codewords c and d in C. Theorem 2.1 If C is a linear code, then d(c) =w(c). Theorem 2.2 A linear code C of dimension k contains precisely 2 k codewords. Lemma 2.2 Let C be a binary linear code of length n. Then, either all the codewords or exactly half of the codewords in C have 0 in the ith coordinate. Proof. Let X be the set of codewords in C with 0 in the ith coordinate and Y be the set of codewords in C with 1 in the ith coordinate. Then, C = X Y. Suppose Y. Let c Y.Sincec+X Y, we find that, X Y.Similarly, c + Y X, andthis 9

10 implies that, Y X. Hence, X = Y. If Y =, C = X has all codewords with 0 in the ith coordinate. Definition. Let C be a binary linear code of length n. The orthogonal code, denoted by C,isgivenby C ={v Z n 2 (v, c) = 0 for all c C}. We call C self-orthogonal if C C and self-dual if C = C. 2.3 Generator martix and parity check martix Another advantage of linear codes is that a binary linear [n, k]-code C can be described simply by giving a basis for C, which consists of k linearly independent codewords in C, rather than having to list all of the 2 k individual codewords in the code. It is customary to put the codewords of a basis for a linear code C into a matrix. Definition. A generator martix for a linear code C is a matrix G whose rows form a basis for C. Definition. A parity-check matrix for a linear code C is a matrix H whose columns form a basis for the dual code C. Theorem 2.3 Matrices G and H are generator and parity-check matrices respectively, for some linear code C if and only if 10

11 i. the rows of G are linearly independent, ii. the columns of H are linearly independent, iii. the number of rows of G plus the number of columns of H equals the number of columns of G which equals the number of rows of H, and iv. GH =0. Theorem 2.4 Let H be a parity-check matrix for a linear code C of length n. Then C has distance d if and only if any set of d 1 rowsofhis linearly independent, and at least one set of d rows of H is linearly dependent. Proof : Suppose v Z n 2. Then, vh is a linear combination of exactly wt(v) rowsofh. If v C and wt(v) =d,thensincevh =0,somedrows of H are linearly dependent. And if vh =0,thenvis a codeword so wt(v) d. 11

12 Chapter 3 Some Construction of Binary Linear Codes 3.1 Motivation We have discussed the properties of binary linear codes. Now, we will like to investigate the different constructions of new codes, and investigate whether they generate optimal binary linear codes. In fact, determining the values of the optimal linear codes has come to known as the main coding theory problem. Unfortunately, very little is currently known about the number that we are looking for. Thus, the introduction of these constructions will hopefully generate good codes from a given set of optimal linear codes. The reason why we are interested in optimal linear codes is because it gives the maximum total transmission rate and error correction rate. It is pointless to have codes with excellent transmission rate but small error correction rate. On the other hand, codes with good error correction rate but poor transmission rate should be 12

13 avoided. Thus, coding theorists are still looking for a really good compromise. Definitions. 1. The transmission rate of a binary linear code C with parameters [n, k, d] is defined to be R(C) =k n. 2. The error correction rate of a binary linear code C with parameters [n, k, d] is defined to be (C) =[(d 1)/2] n. 3. A trivial binary linear code is a linear code with parameter [n, 1,n]or[n, n, 1], for all n N. The [n, 1,n]-linear code is a code with one non-zero codeword, that has all 1 and the weight of this codeword is n. The [n, n, 1]-linear code is a code which contains the whole subspace Z n 2, with a generator matrix row equivalent to an identity matrix, I n, and has minimum weight of Optimal codes are codes with the maximum d, given the length n and the dimension k. 5. Trivial codes are optimal codes. 13

14 3.2 Lengthening a Code Definition Let C be a binary linear code of length n. The process of adding 0 to all codewords in C is referred to as lengthening the code C. Theorem Let C be an [n, k, d] linear code. Then, there exists a linear code C with parameter [n + s, k, d], where s>0. Proof Let G be a generator martix of C. When we add a column of all 0 s to G, we will have a new generator matrix G (1) such that it is a k (n + 1) matrix. Since we add a column of all 0 s, the linear combination of the basis does not affect the minimum weight of C, i.e., d(c) =d. Hence, the parameters of the new code is [n +1,k,d]. By mathematical induction, we are able to obtain new generator matrix G (s) such that G (s) is a (n + s) k matrix. Hence, there exists a linear code C with parameter [n + s, k, d], where s>0. 14

15 Examples Let C = {00000, 11100, 00111, 11011}. Then C = { , , , }. In this case, the length increased is 3. Remarks We may assume that s>0, since when s = 0, the linear code C with parameters [n +0,k,d]=[n, k, d] is already assumed to exist. Please refer to Table 2 on page for the list of optimal codes generated by this construction. 15

16 3.3 Extending a Code Definition Let C be a binary linear code of length n. The code C of length n + 1 obtained from C by adding one extra bit to each codeword in order to make each codeword into a new codeword of even weight is called the extended code of C. In particular, we are interested in codes that have codewords with odd weight initally. If C contains of all codewords that have even weight, extending a code is the same as lengthening the code. Theorem Let C be an [n, k, d]-linear code. Then, given that d is odd, there exists a linear code C with parameter [n +1,k,d+1]. Proof Since C is an [n, k, d]-linear code, there exists c C, wherewt(c) =d(c)=d. Let the extended codeword c be c. Given that d is odd, then by definition, wt(c )=d(c )=wt(c)+1=d+1. 16

17 If C has a k n generator matrix G, then the extended code C has k (n +1) generator matrix G =[G, b], where the last column b of G is appended so that each row of G has even weight. dim(c )=dim(c) =k. Hence, the parameters of C is [n +1,k,d+1]. Examples Let C = {00000, 11100, 00111, 11011}. Then C = {000000, , , }. Remarks Please refer to Table 3 on page for the list of optimal codes generated by this construction. 17

18 3.4 Taking a Subcode from a Code Definition Let C be a binary linear code of length n. The process of deleting a codeword from the basis of C so as to obtain a new code C where the minimum weight of C remains the same, is referred to as taking a subcode of C, orsubcoding the code C. Theorem Let C be an [n, k, d]-linear code. Then, there exists a linear code C with parameter [n, k s, d], where s>0andk>1. Proof Since C is an [n, k, d]-linear code, there exists c C, wherewt(c) =d(c)=d. Let G be a generator matrix of C, wheregis a k n matrix, and contains c in the row basis of C. By deleting any row, r, thathaswt(r) d, r c, from the row basis of G, weobtain C. Since the remaining k 1 rows are still linearly indepentent, dim(c )=k 1. d(c )=wt(c) =d. Hence, the parameters of C is [n, k 1,d]. 18

19 By mathematical induction, there exists a linear code C with parameters [n, k s, d], where s>0. Examples Let C = {0000, 0011, 1100, 1111, 0001, 0010, 1101, 1110} Then, C has a generator matrix G, of the form G = Choosing c = 0001, and deleting the first row of G, we find that C = {0000, 0011, 0001, 0010}. Remarks We may assume that s>0, since when s = 0, the linear code C with parameters [n, k 0,d]=[n, k, d] is already assumed to exist. Also, s<k, for if s k, thesetc becomes an empty set. Please refer to Table 4 on page for the list of optimal codes generated by this construction. 19

20 3.5 Shortening a Code Definition Let C be a binary linear code of length n. A shorten code of C is the set of all codewords of C which are zero at a fixed coordinate with that coordinate deleted. Those codewords in C with 1 at that coordinate shall be removed from C. In particular, we are interested in deleting those coordinates that is not 0 for all codewords in C. If the coordinate is 0 for all codewords in C, we shall use the reverse technique of lengthening, and then use the technique of taking a subcode, which shall be discussed later. Theorem Let C be an [n, k, d]-linear code. Then, there exists a linear code C with parameter [n s, k s, d], where k>s>0. Proof Let G and H be a generator matrix and parity-check matrix of C respectively. Since C is an [n, k, d]-linear code, there exists c C, wherewt(c) =d(c)=d. Suppose c has 0 in the ith coordinate. Case 1 : 20

21 Suppose all codewords in C have 0 in the ith coordinate. Then, by using the reverse technique of lengthening, we obtain C 1 with parameters [n 1,k,d]. By applying subcoding to C 1,withs= 1, we obtained C, with parameter [n 1,k 1,d]. Case 2 : Suppose C has some codewords that has 1 in the ith coordinate. Then, by Lemma 2.2, exactly half of the codewords in C has 1 in the ith coordinate. By deleting the ith coordinate from all the codewords in C, weobtainc. Let the shortened codeword c be c. d(c )=wt(c )=d. C =2 k 2=2 k 1. dim(c )=k 1. Hence, the parameters of C is [n 1,k 1,d]. By mathematical induction, using case 1 and 2 simultaneously, there exists a linear code C with parameter [n s, k s, d], where s>0. Examples Let C = {0000, 1100, 0011, 1111, 1000, 0100, 1011, 0111}. Then, C = {000, 110, 100, 010} is one possible shortened code of C, where in this case, we delete those codewords in C having 1 in the fourth coordinate first, and then removing the fourth coordinate 0, from the remaining codewords. 21

22 Remarks We may assume that s>0, since when s = 0, the linear code C with parameters [n 0,k 0,d]=[n, k, d] is already assumed to exist. Also, s<k, for if s k, thesetc becomes an empty set. Refer to Table 5 on pages for the list of optimal codes generated by this construction. 22

23 3.6 Truncating a Code Definition Let C be a binary linear code of length n. The code obtained by removing a fixed coordinate of C is called a truncated code of C. In particular, we are interested in deleting those coordinates that is not 0 for all codewords in C. Wehavementioned in previous section of the technique of shortening a code with coordinate that is 0 for all codewords in C. Theorem Let C be an [n, k, d]-linear code. Then, there exists a linear code C with parameter [n s, k, d s], where s>0andd>1. Proof Since C is an [n, k, d]-linear code, there exists c C, wherewt(c) =d(c)=d. Suppose c has 1 in the ith coordinate. By deleting the ith coordinate from all the codewords in C, weobtainc. Let the truncated codeword c be c. d(c )=wt(c )=d 1. Also, suppose dim(c ) <k. 23

24 Then, there exists x, y C such that x + c = y, wherewt(c) =1. But, this is a contradiction that d>1. Thus, dim(c )=k. Hence, the parameters of C is [n 1,k,d 1]. By mathematical induction, there exists a linear code C with parameter [n s, k, d s], where s>0. Examples Let C = {0000, 1100, 0011, 1111}. Then, C = {000, 110, 001, 111} is one possible truncated code of C, where in this case, we delete the fourth coordinate of every codewords in C. Also, in this case, the length truncated is, s =1. Remarks We may assume that s>0, since when s = 0, the linear code C with parameters [n 0,k,d 0] = [n, k, d] is already assumed to exist. Also, s<d, for if s d, thesetc becomes an empty set. Often, the technique of shortening and truncating is used simultaneously so as to obtain other optimal codes. Please refer to Table 6 on page for the list of optimal codes generated by this construction. 24

25 3.7 Concatenation of Codes Definition The concatenation of two codewords, c 1 and c 2, is a new string c 1 c 2 formed by writing the elements of c 1 and the elements of c 2 consecutively. The concatenation of two codes, C 1 and C 2, is a set of codewords of the form c 1 c 2 such that c 1 C 1 and c 2 C 2. In particular, the sizes of C 1 and C 2 are both equal. We define concatenation as an injective operation. That is, each codewords in C 1 and C 2 can only be used once in the concatenation. Then, the new code has a generator matrix G =(G 1 G 2 ), where G 1 and G 2 are the generator matrices of C 1 and C 2 respectively. Theorem Let C 1 be an [n 1,k,d 1 ]-linear code and C 2 be an [n 2,k,d 2 ]-linear code. Then, there exists a linear code C with parameter [n 1 + n 2,k,d 1 +d 2 ]. Proof Since C 1 is an [n 1,k,d 1 ]-linear code and C 2 is an [n 2,k,d 2 ]-linear code, there exists c 1 C 1, c 1 0,andc 2 C 2,c 2 0, such that wt(c 1 )=d(c 1 )=d 1 and wt(c 2 )=d(c 2 )=d 2. 25

26 If c 1 =0orc 2 =0,thenC will contain no zero codeword, since 0 is the concatenation of 2 zero codewords. It is a contradication that C is also linear. By concatenating c 1 and c 2,wehavec =c 1 c 2 C. By definition, we concatenate the remaining codewords in C 1 with the remaining codewords in C 2. dim(c )=dim(c 1 )=dim(c 2 )=k. d(c )=wt(c )=wt(c 1 c 2 )=wt(c 1 )+wt(c 2 )=d 1 +d 2. Hence, the parameters of C is [n 1 + n 2,k,d 1 +d 2 ]. Examples Let C 1 = {000, 001, 110, 111} and C 2 = {000, 100, 011, 111}. Then, C = {000000, , , }. Remarks Please refer to Table 7 on page 48 for the list of optimal codes generated by this construction. 26

27 3.8 Combination of Codes Definition The combination of two codes, C 1 and C 2, is a set of codewords of the form c 1 c 2 such that c 1 C 1 and c 2 C 2. We define combination as a surjective operation, which is different from concatenation of two codes. That is, every codewords in C 1 will concatenate with every codewords in C 2. In particular, the sizes of C 1 and C 2 need not to be equal. Theorem Let C 1 be an [n 1,k 1,d 1 ]-linear code and C 2 be an [n 2,k 2,d 2 ]-linear code. Then, there exists a linear code C with parameter [n 1 + n 2,k 1 +k 2, min{d 1,d 2 }]. Proof Since C 1 is an [n 1,k 1,d 1 ]-linear code and C 2 is an [n 2,k 2,d 2 ]-linear code, there exists c 1 C 1, c 1 0,andc 2 C 2,c 2 0, such that wt(c 1 )=d(c 1 )=d 1 and wt(c 2 )=d(c 2 )=d 2. By concatenating c 1 with 0 C 2,wehavewt(c 1)=wt(c 1 )+0=d 1. By concatenating c 2 with 0 C 1,wehavewt(c 2)=wt(c 2 )+0=d 2. 27

28 By definition, we concatenate the remaining codewords in C 1 with the remaining codewords in C 2. C = C 1 C 2 =2 k 1 2 k 2 =2 k 1+k 2. dim(c )=k 1 +k 2. d(c )=min{wt(c 1),wt(c 2)}=min{d 1,d 2 }. Hence, the parameters of C is [n 1 + n 2,k 1 +k 2, min{d 1,d 2 }]. Examples Let C 1 = {00, 01, 10, 11} and C 2 = {00, 11}. Then, C = {0000, 0011, 0100, 0111, 1000, 1011, 1100, 1111}. Remarks Please refer to Table 8 on page 49 for the list of optimal codes generated by this construction. 28

29 3.9 The u(u + v)-construction Theorem Let C 1 be an [n, k 1,d]-linear code and C 2 be an [n, k 2, 2d]-linear code. Let C 1 C 2 be the code consisting of all codewords of the form (u, u + v) =(u 1,u 2,..., u n,u 1 +v 1,u 2 +v 2,..., u n + v n ) with u =(u 1,u 2,..., u n ) C 1 and v =(v 1,v 2,..., v n ) C 2. Then, C 1 C 2 is an [2n, k 1 + k 2, 2d]-linear code. Proof Since C 1 is an [n, k 1,d]-linear code and C 2 is an [n, k 2, 2d]-linear code, there exists c 1 C 1, c 1 0, and c 2 C 2, c 2 0, such that wt(c 1 ) = d(c 1 ) = d and wt(c 2 )=d(c 2 )=2d. Let C 1 C 2 be the set of codewords of the form (u, u+v) defined above, where u C 1, v C 2 and let c C 1 C 2. Case 1 : If u =0andv 0,then min{wt(c )} = min{wt(0,v)}=min{wt(v)} = wt(c 2 )=2d. Case 2 : 29

30 If v =0andu 0,then min{wt(c )} = min{wt(u, u)} = min{wt(u)+wt(u)} = wt(c 1 )+wt(c 1 )=2d. Case 3 : If u 0andv 0,then by the Triangle Inequality, wt(u+v) = wt(u v) wt(u) wt(v), for wt(u) wt(v), wt(c )=wt(u, u + v) = wt(u)+(wt(u) wt(v)) min{wt(v)} = wt(c 2 )=2d. d(c 1 C 2 )=min{wt(c )} =2d. Also, since c is of the form (u, u + v), u has 2 k 1 choices and v has 2 k 2 choices. C 1 C 2 = C 1 C 2 =2 k 1 2 k 2 =2 k 1+k 2. Therefore, dim(c 1 C 2 )=k 1 +k 2. Hence, the parameters of C 1 C 2 is [2n, k 1 + k 2, 2d]. Examples Let C 1 = {00, 10} and C 2 = {00, 11}. Then, C 1 C 2 = {0000, 0011, 1010, 1001} which is a code of length 4, size 4 and minimum distance 2. Remarks Please refer to Table 9 on page 50 for the list of optimal codes generated by this construction. 30

31 3.10 The (G G)-Construction Definition Let u = u 1 u 2...u n Z n 2.Thecomplement of u, denoted by u c, is defined as u c = u c 1 uc 2...uc n such that u c i =1+u i for all i n, i Z 2. Theorem Let C be an [n, k, d]-linear code and G is the generator matrix of C. G G G =, where 0 = (0, 0, 0,..., 0) and 1 = (1, 1, 1,..., 1). 0 1 Let C bethecodewithg as the generator matrix. We define Then, C has parameter [2n, k +1,d ], where d = min{n, 2d} Proof Since C is an [n, k, d]-linear code, there exists c C, c 0, such that wt(c) = d(c)=d. Let C be the set of codewords with a generator matrix G. Let c =(c 1 c 2 ) C,wherec 1 is the string of first n bits and c 2 is the string of the 31

32 next n bits. min{wt(c )} = min{wt(c 1c 2)} = min{wt(c 1 )+wt(c 2 )} = min{wt(c 1 )} + min{wt(c 2 )} = wt(c 1 )+min{wt(c 1 ),wt(c c 1 )} = d+ min{d,n d} = min{d + d,d+ n d} = min{n, 2d}. d(c )=min{wt(c )} = min{n, 2d}. By definition, G contains the basis of C, and thus, the first k rows of G are linearly independent. Since the first n bits of the last row of G are all 0, the last row of G is linearly independent of the first k rows of G. Thus, the (k +1)rowsofG form the basis of C. dim(c )=k+1. Hence, the parameters of C is [2n, k +1, min{n, 2d}]. Examples Let C = {000, 001, 010, 011}. Then, C = {000000, , , , , , , }. Remarks Please refer to Table 10 on page 51 for the list of optimal codes generated by this construction. 32

33 List of Tables In this section, there are 10 tables included. Table 1 lists all the binary linear [n,k]- codes, for n 30, and for all k n, which are optimal. These optimal codes can be obtained from the Brower s table. Table 2 to 10 are the results of the respective constructions. The codes used for these constructions are obtained from Table 1 itself. After computation by the different constructions, the resulting codes are then being compared with Table 1. Table 2 to 10 thus listed all the results that are optimal codes. The range of all the binary linear [n,k]-codes are the same, namely, n 30. Also, in the process of computation, we have omitted those resulting codes that are trival, since we are more interested in consturcting non-trival codes. And, we have omitted those resulting codes that have repeated from one particular construction. For example, in Table 7, the concatenation of [6, 2, 4]-linear code and [6, 2, 4]-linear code and the concatenation of [9, 2, 6]-linear code and [3, 2, 2]-linear code both produce [12, 2, 8]-linear code. The latter one, in our case, will be omitted. 33

34 Table 1 : List of Optimal Binary Linear Codes [n, k, d] [n, k, d] [n, k, d] [n, k, d] [n, k, d] [n, k, d] [n, k, d] [1,1,1] [9,4,4] [13,1,13] [15,13,2] [18,4,8] [20,6,8] [22,4,11] [2,1,2] [9,5,3] [13,2,8] [15,14,2] [18,5,8] [20,7,8] [22,5,10] [2,2,1] [9,6,2] [13,3,7] [15,15,1] [18,6,8] [20,8,8] [22,6,9] [3,1,3] [9,7,2] [13,4,6] [16,1,16] [18,7,7] [20,9,7] [22,7,8] [3,2,2] [9,8,2] [13,5,5] [16,2,10] [18,8,6] [20,10,6] [22,8,8] [3,3,1] [9,9,1] [13,6,4] [16,3,8] [18,9,6] [20,11,5] [22,9,8] [4,1,4] [10,1,10] [13,7,4] [16,4,8] [18,10,4] [20,12,4] [22,10,8] [4,2,2] [10,2,6] [13,8,4] [16,5,8] [18,11,4] [20,13,4] [22,11,7] [4,3,2] [10,3,5] [13,9,3] [16,6,6] [18,12,4] [20,14,4] [22,12,6] [4,4,1] [10,4,4] [13,10,2] [16,7,6] [18,13,3] [20,15,3] [22,13,5] [5,1,5] [10,5,4] [13,11,2] [16,8,5] [18,14,2] [20,16,2] [22,14,4] [5,2,3] [10,6,3] [13,12,2] [16,9,4] [18,15,2] [20,17,2] [22,15,4] [5,3,2] [10,7,2] [13,13,1] [16,10,4] [18,16,2] [20,18,2] [22,16,4] [5,4,2] [10,8,2] [14,1,14] [16,11,4] [18,17,2] [20,19,2] [22,17,3] [5,5,1] [10,9,2] [14,2,9] [16,12,2] [18,18,1] [20,20,1] [22,18,2] [6,1,6] [10,10,1] [14,3,8] [16,13,2] [19,1,19] [21,1,21] [22,19,2] [6,2,4] [11,1,11] [14,4,7] [16,14,2] [19,2,12] [21,2,14] [22,20,2] [6,3,3] [11,2,7] [14,5,6] [16,15,2] [19,3,10] [21,3,12] [22,21,2] [6,4,2] [11,3,6] [14,6,5] [16,16,1] [19,4,9] [21,4,10] [22,22,1] [6,5,2] [11,4,5] [14,7,4] [17,1,17] [19,5,8] [21,5,10] [23,1,23] [6,6,1] [11,5,4] [14,8,4] [17,2,11] [19,6,8] [21,6,8] [23,2,15] [7,1,7] [11,6,4] [14,9,4] [17,3,9] [19,7,8] [21,7,8] [23,3,12] [7,2,4] [11,7,3] [14,10,3] [17,4,8] [19,8,7] [21,8,8] [23,4,12] [7,3,4] [11,8,2] [14,11,2] [17,5,8] [19,9,6] [21,9,8] [23,5,11] [7,4,3] [11,9,2] [14,12,2] [17,6,7] [19,10,5] [21,10,7] [23,6,10] [7,5,2] [11,10,2] [14,13,2] [17,7,6] [19,11,4] [21,11,6] [23,7,9] [7,6,2] [11,11,1] [14,14,1] [17,8,6] [19,12,4] [21,12,5] [23,8,8] [7,7,1] [12,1,12] [15,1,15] [17,9,5] [19,13,4] [21,13,4] [23,9,8] [8,1,8] [12,2,8] [15,2,10] [17,10,4] [19,14,3] [21,14,4] [23,10,8] [8,2,5] [12,3,6] [15,3,8] [17,11,4] [19,15,2] [21,15,4] [23,11,8] [8,3,4] [12,4,6] [15,4,8] [17,12,3] [19,16,2] [21,16,3] [23,12,7] [8,4,4] [12,5,4] [15,5,7] [17,13,2] [19,17,2] [21,17,2] [23,13,6] [8,5,2] [12,6,4] [15,6,6] [17,14,2] [19,18,2] [21,18,2] [23,14,5] [8,6,2] [12,7,4] [15,7,5] [17,15,2] [19,19,1] [21,19,2] [23,15,4] [8,7,2] [12,8,3] [15,8,4] [17,16,2] [20,1,20] [21,20,2] [23,16,4] [8,8,1] [12,9,2] [15,9,4] [17,17,1] [20,2,13] [21,21,1] [23,17,4] [9,1,9] [12,10,2] [15,10,4] [18,1,18] [20,3,11] [22,1,22] [23,18,3] [9,2,6] [12,11,2] [15,11,3] [18,2,12] [20,4,10] [22,2,14] [23,19,2] [9,3,4] [12,12,1] [15,12,2] [18,3,10] [20,5,9] [22,3,12] [23,20,2] 34

35 Table 1 : List of Optimal Binary Linear Codes [n, k, d] [n, k, d] [n, k, d] [n, k, d] [n, k, d] [23,21,2] [25,13,6] [27,1,27] [28,13,8] [29,24,3] [23,22,2] [25,14,6] [27,2,18] [28,14,8] [29,25,2] [23,23,1] [25,15,5] [27,3,15] [28,15,6] [29,26,2] [24,1,24] [25,16,4] [27,4,14] [28,16,6] [29,27,2] [24,2,16] [25,17,4] [27,5,13] [28,17,6] [29,28,2] [24,3,13] [25,18,4] [27,6,12] [28,18,5] [29,29,1] [24,4,12] [25,19,4] [27,7,12] [28,19,4] [30,1,30] [24,5,12] [25,20,3] [27,8,10] [28,20,4] [30,2,20] [24,6,10] [25,21,2] [27,9,10] [28,21,4] [30,3,16] [24,7,10] [25,22,2] [27,10,9] [28,22,4] [30,4,16] [24,8,8] [25,23,2] [27,11,8] [28,23,3] [30,5,15] [24,9,8] [25,24,2] [27,12,8] [28,24,2] [30,6,14] [24,10,8] [25,25,1] [27,13,8] [28,25,2] [30,7,12] [24,11,8] [26,1,26] [27,14,7] [28,26,2] [30,8,12] [24,12,8] [26,2,17] [27,15,6] [28,27,2] [30,9,12] [24,13,6] [26,3,14] [27,16,6] [28,28,1] [30,10,11] [24,14,6] [26,4,13] [27,17,5] [29,1,29] [30,11,10] [24,15,4] [26,5,12] [27,18,4] [29,2,19] [30,12,9] [24,16,4] [26,6,12] [27,19,4] [29,3,16] [30,13,8] [24,17,4] [26,7,11] [27,20,4] [29,4,15] [30,14,8] [24,18,4] [26,8,10] [27,21,4] [29,5,14] [30,15,8] [24,19,3] [26,9,9] [27,22,3] [29,6,13] [30,16,7] [24,20,2] [26,10,8] [27,23,2] [29,7,12] [30,17,6] [24,21,2] [26,11,8] [27,24,2] [29,8,12] [30,18,6] [24,22,2] [26,12,8] [27,25,2] [29,9,11] [30,19,6] [24,23,2] [26,13,7] [27,26,2] [29,10,10] [30,20,5] [24,24,1] [26,14,6] [27,27,1] [29,11,9] [30,21,4] [25,1,25] [26,15,6] [28,1,28] [29,12,8] [30,22,4] [25,2,16] [26,16,5] [28,2,18] [29,13,8] [30,23,4] [25,3,14] [26,17,4] [28,3,16] [29,14,8] [30,24,4] [25,4,12] [26,18,4] [28,4,14] [29,15,7] [30,25,3] [25,5,12] [26,19,4] [28,5,14] [29,16,6] [30,26,2] [25,6,11] [26,20,4] [28,6,12] [29,17,6] [30,27,2] [25,7,10] [26,21,3] [28,7,12] [29,18,6] [30,28,2] [25,8,9] [26,22,2] [28,8,11] [29,19,5] [30,29,2] [25,9,8] [26,23,2] [28,9,10] [29,20,4] [30,30,1] [25,10,8] [26,24,2] [28,10,10] [29,21,4] [25,11,8] [26,25,2] [28,11,8] [29,22,4] [25,12,8] [26,26,1] [28,12,8] [29,23,4] 35

36 Table 2 : Results from Lengthening a Code [n, k, d] s [n + s, k, d] [3,2,2] 1 [4,2,2] [4,3,2] 1 [5,3,2] [5,4,2] 1 [6,4,2] [6,2,4] 1 [7,2,4] [6,5,2] 1 [7,5,2] [7,3,4] 1 [8,3,4] [7,5,2] 1 [8,5,2] [7,6,2] 1 [8,6,2] [8,3,4] 1 [9,3,4] [8,4,4] 1 [9,4,4] [8,6,2] 1 [9,6,2] [8,7,2] 1 [9,7,2] [9,2,6] 1 [10,2,6] [9,4,4] 1 [10,4,4] [9,7,2] 1 [10,7,2] [9,8,2] 1 [10,8,2] [10,5,4] 1 [11,5,4] [10,8,2] 1 [11,8,2] [10,9,2] 1 [11,9,2] [11,3,6] 1 [12,3,6] [11,5,4] 1 [12,5,4] [11,6,4] 1 [12,6,4] [11,9,2] 1 [12,9,2] [11,10,2] 1 [12,10,2] [12,2,8] 1 [13,2,8] [12,4,6] 1 [13,4,6] [12,6,4] 1 [13,6,4] [12,7,4] 1 [13,7,4] [12,10,2] 1 [13,10,2] [12,11,2] 1 [13,11,2] [13,7,4] 1 [14,7,4] [13,8,4] 1 [14,8,4] [13,11,2] 1 [14,11,2] [13,12,2] 1 [14,12,2] [14,3,8] 1 [15,3,8] [14,8,4] 1 [15,8,4] [14,9,4] 1 [15,9,4] [14,12,2] 1 [15,12,2] [14,13,2] 1 [15,13,2] 36

37 Table 2 : Results from Lengthening a Code [n, k, d] s [n + s, k, d] [15,2,10] 1 [16,2,10] [15,3,8] 1 [16,3,8] [15,4,8] 1 [16,4,8] [15,6,6] 1 [16,6,6] [15,9,4] 1 [16,9,4] [15,10,4] 1 [16,10,4] [15,12,2] 1 [16,12,2] [15,13,2] 1 [16,13,2] [15,14,2] 1 [16,14,2] [16,4,8] 1 [17,4,8] [16,5,8] 1 [17,5,8] [16,7,6] 1 [17,7,6] [16,10,4] 1 [17,10,4] [16,11,4] 1 [17,11,4] [16,13,2] 1 [17,13,2] [16,14,2] 1 [17,14,2] [16,15,2] 1 [17,15,2] [17,4,8] 1 [18,4,8] [17,5,8] 1 [18,5,8] [17,8,6] 1 [18,8,6] [17,10,4] 1 [18,10,4] [17,11,4] 1 [18,11,4] [17,14,2] 1 [18,14,2] [17,15,2] 1 [18,15,2] [17,16,2] 1 [18,16,2] [18,2,12] 1 [19,2,12] [18,3,10] 1 [19,3,10] [18,5,8] 1 [19,5,8] [18,6,8] 1 [19,6,8] [18,9,6] 1 [19,9,6] [18,11,4] 1 [19,11,4] [18,12,4,] 1 [19,12,4] [18,15,2] 1 [19,15,2] [18,16,2] 1 [19,16,2] [18,17,2] 1 [19,17,2] [19,6,8] 1 [20,6,8] [19,7,8] 1 [20,7,8] [19,12,4] 1 [20,12,4] [19,13,4] 1 [20,13,4] 37

38 Table 2 : Results from Lengthening a Code [n, k, d] s [n + s, k, d] [19,16,2] 1 [20,16,2] [19,17,2] 1 [20,17,2] [19,18,2] 1 [20,18,2] [20,4,10] 1 [21,4,10] [20,6,8] 1 [21,6,8] [20,7,8] 1 [21,7,8] [20,8,8] 1 [21,8,8] [20,13,4] 1 [21,13,4] [20,14,4] 1 [21,14,4] [20,17,2] 1 [21,17,2] [20,18,2] 1 [21,18,2] [20,19,2] 1 [21,19,2] 38

39 Table 3 : Results from Extending a Code [n, k, d] [n +1,k,d+1] [2,2,1] [3,2,2] [3,3,1] [4,3,2] [4,4,1] [5,4,2] [5,2,3] [6,2,4] [5,5,1] [6,5,2] [6,3,3] [7,3,4] [6,6,1] [7,6,2] [7,4,3] [8,4,4] [7,7,1] [8,7,2] [8,2,5] [9,2,6] [8,8,1] [9,8,2] [9,5,3] [10,5,4] [9,9,1] [10,9,2] [10,3,5] [11,3,6] [10,6,3] [11,6,4] [10,10,1] [11,10,2] [11,2,7] [12,2,8] [11,4,5] [12,4,6] [11,7,3] [12,7,4] [11,11,1] [12,11,2] [12,8,3] [13,8,4] [12,12,1] [13,12,2] [13,3,7] [14,3,8] [13,5,5] [14,5,6] [13,9,3] [14,9,4] [13,13,1] [14,13,2] [14,2,9] [15,2,10] [14,4,7] [15,4,8] [14,6,5] [15,6,6] [14,10,3] [15,10,4] [14,14,1] [15,14,2] [15,5,7] [16,5,8] [15,7,5] [16,7,6] [15,11,3] [16,11,4] [15,15,1] [16,15,2] [16,8,5] [17,8,6] [16,16,1] [17,16,2] [17,2,11] [18,2,12] [17,3,9] [18,3,10] 39

40 Table 3 : Results from Extending a Code [n, k, d] [n +1,k,d+1] [17,6,7] [18,6,8] [17,9,5] [18,9,6] [17,12,3] [18,12,4] [17,17,1] [18,17,2] [18,7,7] [19,7,8] [18,13,3] [19,13,4] [18,18,1] [19,18,2] [19,4,9] [20,4,10] [19,8,7] [20,8,8] [19,10,5] [20,10,6] [19,14,3] [20,14,4] [19,19,1] [20,19,2] [20,2,13] [21,2,14] [20,3,11] [21,3,12] [20,5,9] [21,5,10] [20,9,7] [21,9,8] [20,11,5] [21,11,6] [20,15,3] [21,15,4] [20,20,1] [21,20,2] [21,10,7] [22,10,8] [21,12,5] [22,12,6] [21,16,3] [22,16,4] [21,21,1] [22,21,2] [22,4,11] [23,4,12] [22,6,9] [23,6,10] [22,11,7] [23,11,8] [22,13,5] [23,13,6] [22,17,3] [23,17,4] [22,22,1] [23,22,2] [23,2,15] [24,2,16] [23,5,11] [24,5,12] [23,7,9] [24,7,10] [23,12,7] [24,12,8] [23,14,5] [24,14,6] [23,18,3] [24,18,4] [23,23,1] [24,23,2] [24,3,13] [25,3,14] [24,19,3] [25,19,4] [24,24,1] [25,24,2] 40

41 Table 3 : Results from Extending a Code [n, k, d] [n +1,k,d+1] [25,6,11] [26,6,12] [25,8,9] [26,8,10] [25,15,5] [26,15,6] [25,20,3] [26,20,4] [25,25,1] [26,25,2] [26,2,17] [27,2,18] [26,4,13] [27,4,14] [26,7,11] [27,7,12] [26,9,9] [27,9,10] [26,13,7] [27,13,8] [26,16,5] [27,16,6] [26,21,3] [27,21,4] [26,26,1] [27,26,2] [27,3,15] [28,3,16] [27,5,13] [28,5,14] [27,10,9] [28,10,10] [27,14,7] [28,14,8] [27,17,5] [28,17,6] [27,22,3] [28,22,4] [27,27,1] [28,27,2] [28,8,11] [29,8,12] [28,18,5] [29,18,6] [28,23,3] [29,23,4] [28,28,1] [29,28,2] [29,2,19] [30,2,20] [29,4,15] [30,4,16] [29,6,13] [30,6,14] [29,9,11] [30,9,12] [29,11,9] [30,11,10] [29,15,7] [30,15,8] [29,19,5] [30,19,6] [29,24,3] [30,24,4] [29,29,1] [30,29,2] 41

42 Table 4 : Results from Subcoding [n, k, d] s [n, k s, d] [4,3,2] 1 [4,2,2] [5,4,2] 1 [5,3,2] [6,5,2] 1 [6,4,2] [7,3,4] 1 [7,2,4] [7,6,2] 1 [7,5,2] [8,4,4] 1 [8,3,4] [8,6,2] 1 [8,5,2] [8,7,2] 1 [8,6,2] [9,4,4] 1 [9,3,4] [9,7,2] 1 [9,6,2] [9,8,2] 1 [9,7,2] [10,5,4] 1 [10,4,4] [10,8,2] 1 [10,7,2] [10,9,2] 1 [10,8,2] [11,6,4] 1 [11,5,4] [11,9,2] 1 [11,8,2] [11,10,2] 1 [11,9,2] [12,4,6] 1 [12,3,6] [12,6,4] 1 [12,5,4] [12,7,4] 1 [12,6,4] [12,10,2] 1 [12,9,2] [12,11,2] 1 [12,10,2] [13,7,4] 1 [13,6,4] [13,8,4] 1 [13,7,4] [13,11,2] 1 [13,10,2] [13,12,2] 1 [13,11,2] [14,8,4] 1 [14,7,4] [14,9,4] 1 [14,8,4] [14,12,2] 1 [14,11,2] [14,13,2] 1 [14,12,2] [15,4,8] 1 [15,3,8] [15,9,4] 1 [15,8,4] [15,10,4] 1 [15,9,4] [15,13,2] 1 [15,12,2] [15,14,2] 1 [15,13,2] [16,4,8] 1 [16,3,8] [16,5,8] 1 [16,4,8] [16,7,6] 1 [16,6,6] [16,10,4] 1 [16,9,4] 42

43 Table 4 : Results from Subcoding [n, k, d] s [n, k s, d] [16,11,4] 1 [16,10,4] [16,13,2] 1 [16,12,2] [16,14,2] 1 [16,13,2] [16,15,2] 1 [16,14,2] [17,5,8] 1 [17,4,8] [17,8,6] 1 [17,7,6] [17,11,4] 1 [17,10,4] [17,14,2] 1 [17,13,2] [17,15,2] 1 [17,14,2] [17,16,2] 1 [17,15,2] [18,5,8] 1 [18,4,8] [18,6,8] 1 [18,5,8] [18,9,6] 1 [18,8,6] [18,11,4] 1 [18,10,4] [18,12,4] 1 [18,11,4] [18,15,2] 1 [18,14,2] 43

44 Table 5 : Results from Shortening a Code [n, k, d] s [n s, k s, d] [4,3,2] 1 [3,2,2] [5,3,2] 1 [4,2,2] [5,4,2] 1 [4,3,2] [6,3,3] 1 [5,2,3] [6,4,2] 1 [5,3,2] [6,5,2] 1 [5,4,2] [7,3,4] 1 [6,2,4] [7,4,3] 1 [6,3,3] [7,5,2] 1 [6,4,2] [7,6,2] 1 [6,5,2] [8,3,4] 1 [7,2,4] [8,4,4] 1 [7,3,4] [8,6,2] 1 [7,5,2] [8,7,2] 1 [7,6,2] [9,4,4] 1 [8,3,4] [9,6,2] 1 [8,5,2] [9,7,2] 1 [8,6,2] [9,8,2] 1 [8,7,2] [10,4,4] 1 [9,3,4] [10,5,4] 1 [9,4,4] [10,6,3] 1 [9,5,3] [10,7,2] 1 [9,6,2] [10,8,2] 1 [9,7,2] [10,9,2] 1 [9,8,2] [11,3,6] 1 [10,2,6] [11,4,5] 1 [10,3,5] [11,5,4] 1 [10,4,4] [11,6,4] 1 [10,5,4] [11,7,3] 1 [10,6,3] [11,8,2] 1 [10,7,2] [11,9,2] 1 [10,8,2] [11,10,2] 1 [10,9,2] [12,4,6] 1 [11,3,6] [12,6,4] 1 [11,5,4] [12,7,4] 1 [11,6,4] [12,8,3] 1 [11,7,3] [12,9,2] 1 [11,8,2] [12,10,2] 1 [11,9,2] [12,11,2] 1 [11,10,2] 44

45 Table 5 : Results from Shortening a Code [n, k, d] s [n s, k s, d] [13,4,6] 1 [12,3,6] [13,6,4] 1 [12,5,4] [13,7,4] 1 [12,6,4] [13,8,4] 1 [12,7,4] [13,9,3] 1 [12,8,3] [13,10,2] 1 [12,9,2] [13,11,2] 1 [12,10,2] [13,12,2] 1 [12,11,2] [14,3,8] 1 [13,2,8] [14,4,7] 1 [13,3,7] [14,5,6] 1 [13,4,6] [14,6,5] 1 [13,5,5] [14,7,4] 1 [13,6,4] [14,8,4] 1 [13,7,4] [14,9,4] 1 [13,8,4] [14,10,3] 1 [13,9,3] [14,11,2] 1 [13,10,2] [14,12,2] 1 [13,11,2] [14,13,2] 1 [13,12,2] [15,4,8] 1 [14,3,8] [15,5,7] 1 [14,4,7] [15,6,6] 1 [14,5,6] [15,7,5] 1 [14,6,5] [15,8,4] 1 [14,7,4] [15,9,4] 1 [14,8,4] [15,10,4] 1 [14,9,4] [15,11,3] 1 [14,10,3] [15,12,2] 1 [14,11,2] [15,13,2] 1 [14,12,2] [15,14,2] 1 [14,13,2] [16,4,8] 1 [15,3,8] [16,5,8] 1 [15,4,8] [16,7,6] 1 [15,6,6] [16,8,5] 1 [15,7,5] [16,9,4] 1 [15,8,4] [16,10,4] 1 [15,9,4] 45

46 Table 6 : Results from Truncating a Code [n, k, d] s [n s, k, d s] [5,2,3] 1 [4,2,2] [6,2,4] 1 [5,2,3] [6,3,3] 1 [5,3,2] [7,3,4] 1 [6,3,3] [7,4,3] 1 [6,4,2] [8,2,5] 1 [7,2,4] [8,4,4] 1 [7,4,3] [9,2,6] 1 [8,2,5] [9,5,3] 1 [8,5,2] [10,3,5] 1 [9,3,4] [10,5,4] 1 [9,5,3] [10,6,3] 1 [9,6,2] [11,2,7] 1 [10,2,6] [11,3,6] 1 [10,3,5] [11,4,5] 1 [10,4,4] [11,6,4] 1 [10,6,3] [11,7,3] 1 [10,7,2] [12,2,8] 1 [11,2,7] [12,4,6] 1 [11,4,5] [12,7,4] 1 [11,7,3] [12,8,3] 1 [11,8,2] [13,3,7] 1 [12,3,6] [13,5,5] 1 [12,5,4] [13,8,4] 1 [12,8,3] [13,9,3] 1 [12,9,2] [14,2,9] 1 [13,2,8] [14,3,8] 1 [13,3,7] [14,4,7] 1 [13,4,6] [14,5,6] 1 [13,5,5] [14,6,5] 1 [13,6,4] [14,9,4] 1 [13,9,3] [14,10,3] 1 [13,10,2] [15,2,10] 1 [14,2,9] [15,4,8] 1 [14,4,7] [15,5,7] 1 [14,5,6] [15,6,6] 1 [14,6,5] [15,7,5] 1 [14,7,4] [15,10,4] 1 [14,10,3] [15,11,3] 1 [14,11,2] 46

47 Table 6 : Results from Truncating a Code [n, k, d] s [n s, k, d s] [16,5,8] 1 [15,5,7] [16,7,6] 1 [15,7,5] [16,8,5] 1 [15,8,4] [16,11,4] 1 [15,11,3] [17,2,11] 1 [16,2,10] [17,3,9] 1 [16,3,8] [17,6,7] 1 [16,6,6] [17,8,6] 1 [16,8,5] [17,9,5] 1 [16,9,4] [17,12,3] 1 [16,12,2] [18,2,12] 1 [17,2,11] [18,3,10] 1 [17,3,9] [18,6,8] 1 [17,6,7] [18,7,7] 1 [17,7,6] [18,9,6] 1 [17,9,5] [18,12,4] 1 [17,12,3] [18,13,3] 1 [17,13,2] [19,4,9] 1 [18,4,8] [19,7,8] 1 [18,7,7] [19,8,7] 1 [18,8,6] [19,10,5] 1 [18,10,4] [19,13,4] 1 [18,13,3] [19,14,3] 1 [18,14,2] [20,2,13] 1 [19,2,12] [20,3,11] 1 [19,3,10] [20,4,10] 1 [19,4,9] [20,5,9] 1 [19,5,8] [20,8,8] 1 [19,8,7] [20,9,7] 1 [19,9,6] [20,10,6] 1 [19,10,5] [20,11,5] 1 [19,11,4] [20,14,4] 1 [19,14,3] [20,15,3] 1 [19,15,2] [21,2,14] 1 [20,2,13] [21,3,12] 1 [20,3,11] 47

48 Table 7 : Results from Concatenation of Codes [n 1,k,d 1 ] [n 2,k,d 2 ] [n 1 +n 2,k,d 1 +d 2 ] [2,2,1] [2,2,1] [4,2,2] [3,2,2] [2,2,1] [5,2,3] [3,2,2] [3,2,2] [6,2,4] [4,2,2] [3,2,2] [7,2,4] [4,3,2] [4,3,2] [8,3,4] [5,2,3] [3,2,2] [8,2,5] [5,2,3] [5,2,3] [10,2,6] [5,3,2] [4,3,2] [9,3,4] [5,4,2] [5,4,2] [10,4,4] [6,2,4] [3,2,2] [9,2,6] [6,2,4] [5,2,3] [11,2,7] [6,2,4] [6,2,4] [12,2,8] [6,3,3] [4,3,2] [10,3,5] [6,3,3] [6,3,3] [12,3,6] [6,5,2] [6,5,2] [12,5,4] [7,2,4] [6,2,4] [13,2,8] [7,3,4] [4,3,2] [11,3,6] [7,3,4] [6,3,3] [13,3,7] [7,3,4] [7,3,4] [14,3,8] [8,2,5] [6,2,4] [14,2,9] [8,2,5] [8,2,5] [16,2,10] [8,3,4] [7,3,4] [15,3,8] [8,3,4] [8,3,4] [16,3,8] [8,4,4] [5,4,2] [13,4,6] [8,4,4] [8,4,4] [16,4,8] [9,2,6] [6,2,4] [15,2,10] [9,2,6] [8,2,5] [17,2,11] [9,2,6] [9,2,6] [18,2,12] [9,3,4] [7,3,4] [16,3,8] [9,4,4] [8,4,4] [17,4,8] [9,4,4] [9,4,4] [18,4,8] [10,2,6] [9,2,6] [19,2,12] [10,3,5] [7,3,4] [17,3,9] [11,2,7] [9,2,6] [20,2,13] [11,2,7] [11,2,7] [22,2,14] 48

49 Table 8 : Results from Combination of Codes [n 1,k 1,d 1 ] [n 2,k 2,d 2 ] [n 1 +n 2,k 1 +k 2, min{d 1,d 2 }] [2,1,2] [2,1,2] [4,2,2] [3,2,2] [2,1,2] [5,3,2] [3,2,2] [3,2,2] [6,4,2] [4,3,2] [3,2,2] [7,5,2] [4,3,2] [4,2,2] [8,5,2] [4,3,2] [4,3,2] [8,6,2] [5,3,2] [4,3,2] [9,6,2] [5,4,2] [4,3,2] [9,7,2] [5,4,2] [5,3,2] [10,7,2] [5,4,2] [5,4,2] [10,8,2] [6,4,2] [5,4,2] [11,8,2] [6,5,2] [5,4,2] [11,9,2] [6,5,2] [6,4,2] [12,9,2] [6,5,2] [6,5,2] [12,10,2] [7,5,2] [6,5,2] [13,10,2] [7,6,2] [6,5,2] [13,11,2] [7,6,2] [7,5,2] [14,11,2] [7,6,2] [7,6,2] [14,12,2] [8,4,4] [4,1,4] [12,5,4] [8,6,2] [7,6,2] [15,12,2] [8,6,2] [8,6,2] [16,12,2] [8,7,2] [7,6,2] [15,13,2] [8,7,2] [8,6,2] [16,13,2] [8,7,2] [8,7,2] [16,14,2] 49

50 Table 9 : Results from The u(u + v)-construction [n, k 1,d] [n, k 2, 2d] [2n, k 1 + k 2, 2d] [2,2,1] [2,1,2] [4,3,2] [3,3,1] [3,2,2] [6,5,2] [4,2,2] [4,1,4] [8,3,4] [4,3,2] [4,1,4] [8,4,4] [4,4,1] [4,2,2] [8,6,2] [4,4,1] [4,3,2] [8,7,2] [5,5,1] [5,3,2] [10,8,2] [5,5,1] [5,4,2] [10,9,2] [6,3,3] [6,1,6] [12,4,6] [6,4,2] [6,2,4] [12,6,4] [6,5,2] [6,2,4] [12,7,4] [6,6,1] [6,4,2] [12,10,2] [6,6,1] [6,5,2] [12,11,2] [7,5,2] [7,2,4] [14,7,4] [7,5,2] [7,3,4] [14,8,4] [7,6,2] [7,3,4] [14,9,4] [7,7,1] [7,5,2] [14,12,2] [7,7,1] [7,6,2] [14,13,2] [8,3,4] [8,1,8] [16,4,8] [8,4,4] [8,1,8] [16,5,8] [8,5,2] [8,4,4] [16,9,4] [8,6,2] [8,4,4] [16,10,4] [8,7,2] [8,4,4] [16,11,4] [8,8,1] [8,5,2] [16,13,2] [8,8,1] [8,6,2] [16,14,2] [8,8,1] [8,7,2] [16,15,2] [9,6,2] [9,4,4] [18,10,4] [9,7,2] [9,4,4] [18,11,4] [9,8,2] [9,4,4] [18,12,4] [9,9,1] [9,6,2] [18,15,2] [9,9,1] [9,7,2] [18,16,2] [9,9,1] [9,8,2] [18,17,2] [10,3,5] [10,1,10] [20,4,10] [10,7,2] [10,5,4] [20,12,4] [10,8,2] [10,5,4] [20,13,4] [10,9,2] [10,5,4] [20,14,4] [10,10,1] [10,7,2] [20,17,2] [10,10,1] [10,8,2] [20,18,2] 50

51 Table 10 : Results from The (G G)-Construction [n, k, d] d = min{n, 2d} [2n, k +1,d ] [2,1,2] 2 [4,2,2] [2,2,1] 2 [4,3,2] [3,2,2] 3 [6,3,3] [3,3,1] 2 [6,4,2] [4,2,2] 4 [8,3,4] [4,3,2] 4 [8,4,4] [4,4,1] 2 [8,5,2] [5,2,3] 5 [10,3,5] [5,3,2] 4 [10,4,4] [5,4,2] 4 [10,5,4] [6,2,4] 6 [12,3,6] [6,3,3] 6 [12,4,6] [6,4,2] 4 [12,5,4] [6,5,2] 4 [12,6,4] [7,3,4] 7 [14,4,7] [7,4,3] 6 [14,5,6] [7,6,2] 4 [14,7,4] [8,2,5] 8 [16,3,8] [8,3,4] 8 [16,4,8] [8,4,4] 8 [16,5,8] [9,3,4] 8 [18,4,8] [9,4,4] 8 [18,5,8] [10,3,5] 10 [20,4,10] [10,5,4] 8 [20,6,8] [11,3,6] 11 [22,4,11] [11,4,5] 10 [22,5,10] [11,6,4] 8 [22,7,8] [12,3,6] 12 [24,4,12] [12,4,6] 12 [24,5,12] [12,7,4] 8 [24,8,8] [13,3,7] 13 [26,4,13] [13,4,6] 12 [26,5,12] [14,3,8] 14 [28,4,14] [14,4,7] 14 [28,5,14] [14,5,6] 12 [28,6,12] [15,4,8] 15 [30,5,15] [15,5,7] 14 [30,6,14] [15,6,6] 12 [30,7,12] 51

52 Conclusion In general, the binary linear codes for small n, in our cases, n 30, and all k, have its optimality. In all these cases, the upper bound of d is equal to the lower bound of d. But, for big n, n>30, the maximum d found in many cases are less than the upper bound d. It is of great challenge to discover these new optimal codes. There will be even a greater discovery if we are able to develop an algorithm to find optimal linear codes. Also, those binary linear codes that has not reached its optimality, are often those codes with an odd upper bound. And, often then not, the lower bound, that was found is often an even number. One of the complexity is also due that we have so far given many constructions that can generate good codes that have a maximum even d. Construction such as extending, shortening, truncating and the u(u+v)-construction often generate good codes that produces even d, Thus, it is of great interest to construct good codes that may end with a maximum odd d. 52

53 Bibliography [1] E.F. Assmus Jr, J.D. Key, Designs and Their Codes, Cambridge, [2] John Baylis, Error-Correcting Codes : A Mathematical Introduction, Chapman & Hall, [3] Brower s table, Website address aeb/voorlincod.html [4] D. G. Hoffman, D. A. Leonard, C. C. Lindner, K. T. Phelps, C. A. Rodger, J. R. Wall, Coding Theory : The Essentials, Marcel Dekker, Inc., [5] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Vol. I, North-Holland, [6] Vera Pless, Introduction to the Theory of Error-Correcting Codes, John Wiley & Sons, Inc., [7] Steven Roman, Introduction to Coding and Information Theory, Springer,