5-48 1 On Construction of the (24, 12, 8) Golay Codes Xiao-Hong Peng, Member , IEEE , and Paddy G. Farrell, Life Fellow, IEEE through the extension [9]; ii) Cu is the (8, 4, 4) first-order Abstract —Two product array codes are used to construct the Reed-Muller code and C is formed by reversing the (24, 12, 8) binary Golay code through the direct sum operation. v This construction provides a systematic way to find proper (8, 4, codewords of Cu except for the overall parity check [10]; and 4) linear block component codes for generating the Golay code, iii) C is also the (8, 4, 4) first-order Reed-Muller code and and it generates and extends previously existing methods that use u a similar construction framework. The code constructed is simple Cv is a column permutation of Cu [11]. to decode. In this paper, we present a construction of the (24, 12, 8) Golay code based on two array codes. In this construction, Index Terms —Array codes, block codes, code construction, four component codes are involved: a (3, 2, 2) single-parity- direct sum, Golay codes. check (SPC) code, a (3, 1, 3) repetition code and two (8, 4, 4) linear block codes. We have discovered that given an (8, 4, 4) I. INTRODUCTION code in systematic form, there exist eight other different (8, 4, 4) codes obtained either through proper row permutation on HE (24, 12, 8) binary block code, denoted by C , was 24 the parity submatrix of the generator matrix of the first (8, 4, Toriginally constructed by extending the (23, 12, 7) Golay 4) code, or by applying a set of construction rules. These nine code [1], a unique 3-error correcting perfect code. Because of (8, 4, 4) codes (the original plus the eight others) are of the the optimality and attractive structure of C24 which is self dual same isomorphism type (with the same length, dimension and and doubly even [2], it has received considerable attention, weight distribution), but represent different code subspaces. leading to a large number of construction methods. Using Using the given (8, 4, 4) code, together with any one of the eight (8, 4, 4) codes obtained, in all cases leads to the design theory, C24 can be formed through a 2-(11, 6, 3) design or the 5-(24, 8, 1) Steiner system [3]. All the codewords of construction of the (24, 12, 8) Golay code. This construction C can also be generated by ordering a set of words of length systematizes and extends our previous [12] and other existing 24 methods, including [2], [9]-[11], that apply the 24 over the {0, 1} alphabet lexicographically [4]. Other + + + + approaches based on constructions in a larger field include the a x b x a b x framework and use two (8, 4, 4) codes. use of a Reed-Solomon code over F8 [5], the hexacode over The code constructed can be, as in the case of related constructions [2], [11], decoded with low complexity. F4 [6] and the cubic residue code over F4 [7] or the Mathieu group M 24 [8]. C24 can also be constructed using component codes with shorter length and smaller dimension. This method II. CONSTRUCTION METHOD is based on the so-called Turyn or a + x b + x a + b + x A. The Generator matrices construction [9], where a,b ∈ C , x ∈ C , and C and C are u v u v The two array codes concerned are both two-dimensional two component codes with the same length. There are three product codes. A product code C is formed by a direct product examples utilizing this construction: i) Cu is the extended (7, = [2] of two component codes C1 (n1 , k1 , d 1 ) and 4, 3) cyclic Hamming code and Cv is formed by reversing the = C 2 (n2 , k 2 , d 2 ) . The generator matrix, G, of C is codewords of Cu except for the overall parity check added represented in the form of a Kronecker product (denoted by ⊗ ) of generator matrices of its component codes, G1 and G2 , i.e., Manuscript received January 19, 2005; revised July 7, 2005 and December 15, 2005, respectively. = ⊗ = ( )1( ) = ⊗ = ( )2( ) X.-H. Peng is with the Electronic Engineering Subject Group, School of G G1 G2 g i, j G2 or G G2 G1 g i, j G1 (1) Engineering & Applied Science, Aston University, Birmingham B4 7ET, UK. (phone: +44 (0)121 2043527; fax: +44 (0)121 2043682; e-mail: x-h.peng@ aston.ac.uk). = ( )1( ) = ( )2( ) where G1 g i, j and G2 g i, j . The resulting code is an P. G. Farrell is Visiting Professor in the Department of Communication Systems, Lancaster University, Lancaster LA1 4WA, UK. He can be reached (n1n2 , k1k 2 , d1d 2 ) product code, and G is of size at 7 The Drive, Deal, Kent, CT14 9AE, UK; tel +44 (0)1304 363532; email (k k ) × (n n ) . [email protected]. 1 2 1 2 5-48 2 The first array code C is the (24, 8, 8) product code elements of the codewords of C A always correspond in = × constituted by C C1 C2 , where C1 and C2 denote an (8, 4, position to the zero elements of the codewords of CB , and vice 4) linear systematic block code and a (3, 2, 2) single-parity- versa. This means that C A and CB are guaranteed to be check code, respectively. The generator matrix of C is 2 disjoint. ′ G = 1 0 1 Lemma 1: Code subspaces C and C defined in (2) and (3) 2 0 1 1 are disjoint. Proof : We need to show that for any α and α ′ defined in Therefore, according to (1), the generator matrix of C is given (4), α + α ′ = 0 implies that α = 0 and α ′ = 0 . Codeword α by can be expressed as ( a, 0, a), (0, b, b) or ( a, b, a+b), where ∈ α ′ a,b C1 , according (2). Also, can be expressed as ( x, x, G 0 G = ⊗ = 1 1 ∈ ′ α + α ′ = G G2 G1 (2) x), where x C1 , according to (3). Suppose that 0 . 0 G1 G1 Then we have where G is the 4 × 8 generator matrix of C , and ‘ 0’ 1 1 (a + x, x, a + x) = 0 )i( represents a 4 × 8 null matrix. It is noted that the (24, 8, 8) α + α ′ = or (x, b + x, b + x) = 0 )ii( code is also used in [13] for constructing the (24, 12, 8) code. + + + + = However, the construction presented in this reference leads to or (a x, b x, a b x) 0 iii( ) a nonlinear (24, 12) code, since one of the other component codes, the (8, 3) code, used in the construction is nonlinear. From (iii), it implies that a + x = ,0 b + x = 0 and No direct proof is given for showing that the distance of the a + b + x = 0 . Thus it is easy to conclude that a = 0 , b = 0 code is 8, although its weight distribution turns out to be that and x = 0 , leading to α = 0 and α ′ = 0 . The same result can of the extended Golay code. be attained for (i) and (ii). The second array code C′ is the (24, 4, 12) product code □ constituted by C′ = C′ × C′ , where C′ is one of the 8 other 1 2 1 Following Lemma 1, we now can regard code Cˆ as a result different (8, 4, 4) codes and C′ a (3, 1, 3) repetition code with 2 of the direct sum, denoted by ⊕ , of C and C′ , i.e., the generator matrix Cˆ = C ⊕ C′ (5) G′ = (1 1 1) . 2 ˆ = + ′ Thus the generator matrix of C′ is given by with the dimension: dim (C) dim (C) dim (C ) = 8 + 4 =12. The generator matrix of Cˆ is therefore given by ′ = ′ ⊗ ′ = ( ′ ′ ′) G G2 G1 G1 G1 G1 (3) G 0 G G 1 1 ′ × ′ Gˆ = = 0 G G (6) where G1 is the 4 8 generator matrix of C1 , and obtained 1 1 G′ ′ ′ ′ G1 G1 G1 from G1 , as described later. C and C′ can be regarded as code subspaces of the vector ˆ ˆ = + ′ We can see from (6) that C is an ( n, k) = (24, 12) linear code. space V24 . Let C C C be the code subspace of length 24 We also notice that the codewords in Cˆ are of the form spanned by C and C′ . Each codeword αˆ in Cˆ can be + + + + ∈ ∈ ′ expressed as a sum a x b x a b x , where a,b C1 and x C , which is also the case for the Turyn [9] or cubing [11] construction. αˆ = α + α ′ α ∈ C and α ′ ∈ C′. (4) The minimum distance of Cˆ dependents on the structures of C and C′ , or G and G′ . We present next a method of If C and C′ are disjoint, or, in other words, all the row vectors 1 1 1 1 obtaining G and G′ which guarantees that the minimum of G and G′ are linearly independent, the new code subspace 1 1 ˆ Cˆ can be simply referred as the direct sum of C and C′ . Note distance of C is 8. Let C be an (8, 4, 4) systematic code. The generator that the direct sum construction or | u|v|-construction [2] is a 1 special case of the general direct sum operation adopted here. matrix of C1 is expressed as In that case, the two codes involved, e.g.