A new asymptotically optimal code for the positive integers

A method of constructing binary linear codes is presentedthat have a minimum Hamming distance of five. Some efficient newcodes are obtained. In particular, a (47,36,5) code is obtained that hassix more information bits than the best previously known code with 11check bits.In Section II, we propose a general formulation of paritycheck matrix for the desired code. We then derive necessaryconditions for the code to have a minimum distance of 5. InSection III, we describe some new codes that satisfy the neces-sary conditions derived in Section II. In Section IV, we describethe construction of two additional new codes and make someremarks on the construction of long binary linear codes with aminimum distance of five. Index Terms -Error-correcting code, linear block code.II. FORMULATION OF PARITY-CHECK MATRIX I. INTR~DU~IONBinary linear codes with a minimum Hamming distance of fivecan be constructed from the well known BCH theory [l]-[3]. LetA be the minimum distance, r the number of check bits, and nthe length of the code. For a given m, a primitive BCH codewith d = 5, r = 2m, andn = 2” 1 can be constructed. Moreefficient codes can be constructed from nonprimitive cycliccodes. Let II = 2” + 1, let /3 be a primitive root of X” 1, and C Manuscript received June 12, 1990; revised March 5, 1991.The author is with the IBM Corporation, P.O. Box 950, DepartmentD18, Building 707, Poughkeepsie, NY 12602.IEEE Log Number 9101398.the binary cyclic code of length II with 2m check bits, generatedby the minimum polynomial of p. Then &I;’: i = 1,2,2”-l,2”-’ + 1,2” 1,2”f forms a subset of the roots of the genera-tor polynomial of C. If m is even, the minimum distance of C isat least equal to 5 by HT bound [4]. If m is odd, let C’ be thecode that consists of the set of all even weight codewords of C.Code C’ has 2m + 1 check bits and has a minimum distanced 2 6. The code can be punctured by deleting one of the checkbits to yield a code of length 2” with 2m check bits and d 2 5.Thus, a binary cyclic code with n = 2m + 1 for m even, andn = 2” for m odd that has d 2 5 and r = 2m can always beconstructed. These codes have one or two more information bitsthan the primitive BCH codes with the same number of checkbits.Binary linear codes with d = 5 and an odd number of checkbits can be constructed by shortening BCH codes. Let t be theweight of a minimum weight nonzero vector I’ of the dual codeof a BCH code of length n with d = 5 and r = 2m. The subcodeof the BCH code whose code words have zeros at the t positionscorresponding to the ones of I/ has an effective codelength ofn t with r = 2m 1 and d 2 5. Using this shortening tech-nique, a binary linear code of minimum distance 5 with n =2” +21(m+1)/21 -1 can be constructed [3], where lx] is theinteger part of x.In this correspondence, we present another technique forconstructing binary linear codes of minimum distance 5. A codewith a minimum distance d, codelength n and the number ofinformation bits k will be called an (n, k, d) code. Our approachis to extend the length of a known (n, k,5) code by adding oneadditional check bit. In some cases this construction yields codesthat are equal to or better than the best previously known codeswith d = 5. In particular, a (47,36,5) code is constructed thathas 6 more information bits than the (41,30,5) code, which isthe best code compiled in [5] with r = 11 and d = 5. We alsoconstruct (33,23,5) and (79,66,5> codes.] An (n, k, d) binary linear code is a k-dimensional subspace ofan n-dimensional binary vector space [l]-[3]. The dimensionr = r k of the null space of the code is the number of checkbits. An (n, k, d) code can be specified by an r X n binary paritycheck matrix. The vector space formed by all possible combina-tions of the row vectors of the parity check matrix is the nullspace of the code, i.e., the dual code. In this correspondence, wespecify a linear code by its parity check matrix.Let B and Q be two binary matrices with r rows and n1,n2columns, respectively. We define a binary linear code C of ‘J. B. Shearer has obtained a (82,68,6) code among other efficientcodes through a computer search [6]. OOlS-9448/91/0900-1429$01.00 01991 IEEE