Weight enumerators of self-dual codes

Some construction techniques for self-dual codes are investigated, and the authors construct a singly-even self-dual [48,24,10]-code with a weight enumerator that was not known to be attainable. It is shown that there exists a singly-even self-dual code C' of length n =48 and minimum weight d=10 whose weight enumerator is prescribed in the work of J.H. Conway et al. (see ibid., vol.36, no.5, p.1319-33, 1990). Two self-dual codes of length n are called neighbors, provided their intersection is a code of dimension (n/2)-1. The code C' is a neighbor of the extended quadratic residue code of length 48     

Shadow codes and weight enumerators

The technique of using shadow codes to build larger self-dual codes is extended to codes over arbitrary fields. It is shown how to build the codes and how to determine the new weight enumerator as well. For codes over fields equipped with a square root of -1 and not of characteristic 2, a self-dual code of length n+2 can be built from a self-dual code of length n; for codes over a field without a square root of -1 and not of characteristic 2 a self-dual code of length n+4 is built from a self-dual code of length n; and for codes over fields of characteristic 2 the length of the new self-dual code depends on the presence of the all-one vector in the subcode chosen. In certain cases using the subcode of vectors orthogonal to the all-one vector, the new weight enumerator can be calculated directly from the original weight enumerator. Specific examples of the technique are illustrated for codes over F₃, F₄, and F₅     

The third support weight enumerators of the doubly-even, self-dual [32,16,8] codes

We present combinatorial methods for computing the third support weight enumerators of the five doubly-even, self-dual [32,16,8] codes. The methods exploit relationships that exist between support weight enumerators and complete coset weight enumerators of a self-dual code.     

On asymptotic ensemble weight enumerators of LDPC-like codes

For LDPC-like codes such as LDPC, GLDPC, and DGLDPC codes, it is well known that the error floor can be caused by the codewords of small weights or stopping sets of small sizes. In this paper, we investigate the computation of asymptotic weight enumerators such that it becomes a convenient tool to determine a good distribution of code ensembles. In addition, by analyzing the first order approximation, we derive a condition to obtain a negative asymptotic growth rate of the codewords of small linear-sized weights, which is an important constraint for distribution optimization. Also the weight enumerators of turbo and repeat-accumulate codes are investigated. Furthermore, we extend our results to nonbinary DGLDPC codes. Generalization to N-layer and convolutional code based LDPC-like codes are also developed.     

On the computation of weight enumerators for convolutional codes

Performance bounds for maximum-likelihood decoding of convolutional codes over memoryless channels are commonly measured using the distance weight enumerator T(x,y), also referred to as the transfer function, of the code. This paper presents an efficient iterative method to obtain T(x,y) called the state reduction algorithm. The algorithm is a systematic technique to simplify signal flow graphs that algebraically manipulate the symbolic adjacency matrix associated with the convolutional code. Next, the algorithm is modified to compute the first few terms of the series expansion of T(1,y) and {/spl part/T(x,y)//spl part/x}/sub x=1/ (the distance spectra) without first computing the complete T(x,y).     

On the input-output weight enumerators of product accumulate codes

In this letter, a generalized MacWilliams transform that relates the input-redundancy weight enumerator of a systematic binary linear block code to that of its dual code is first presented. Based on this transform, the input-output weight enumerators of direct-product single-parity-check codes and the type-II product-accumulate codes are then derived, and used to analyze the asymptotic bit error performance of these codes.     

Quantum weight enumerators

In a recent paper, Shor and Laflamme (see Phys. Rev. Lett., vol.78, p.1600-2, 1997) defined two “weight enumerators” for quantum error-correcting codes, connected by a MacWilliams transform, and used them to give a linear programming bound for quantum codes. We introduce two new enumerators which, while much less powerful at producing bounds, are useful tools nonetheless. The new enumerators are connected by a much simpler duality transform, clarifying the duality between Shor and Laflamme's enumerators. We also use the new enumerators to give a simpler condition for a quantum code to have specified minimum distance, and to extend the enumerator theory to codes with block size greater than 2     

卡氏积码的MDR码和自对偶码

定义了Z_(r_1),Z_(r_2)…,Z_(r_s)上线性码C_1,C_2,…,C_s的卡氏积码.利用子模同构定理,研究了在Z_(r_1)×Z_(r_2)×…×Z_(r_s)上卡氏积码C_1×C_2×…×C_s的秩与在Z_(r_1),Z_(r_2),…,Z_(r_s)码C_1,C_2,…,C_s的秩的关系,借助这一关系,得到了MDR码的卡氏积仍为MDR码和自对偶码的卡氏积码也为自对偶码.     

Cubic self-dual binary codes

We study binary self-dual codes with a fixed point free automorphism of order three. All binary codes of that type can be obtained by a cubic construction that generalizes Turyn's. We regard such "cubic" codes of length 3/spl lscr/ as codes of length /spl lscr/ over the ring F/sub 2//spl times/F/sub 4/. Classical notions of Type II codes, shadow codes, and weight enumerators are adapted to that ring. Two infinite families of cubic codes are introduced. New extremal binary codes in lengths /spl les/ 66 are constructed by a randomized algorithm. Necessary conditions for the existence of a cubic [72,36,16] Type II code are derived.     

Gleason's theorem on self-dual codes

The weight enumerator of a code is the polynomial begin{equation} W(x,y)= sum_{r=0}^n A_r x^{n-r} y^r, end{equation} wherendenotes the block length andA_r, denotes the number of codewords of weightr. LetCbe a self-dual code overGF(q)in which every weight is divisible byc. Then Gleason's theorem states that 1) ifq= 2 andc= 2, the weight enumerator ofCis a sum of products of the polynomialsx^2 + y^2andx^2y^2 (x^2 - y^2 )^2ifq= 2 andc= 4, the weight enumerator is a sum of products ofx^8 + 14x^4 y^4 + y^8andx^4 y^4 (x^4 - y^4)^4; and 3) ifq= 3 andc= 3, the weight enumerator is a sum of products ofx^4 + 8xy^3andy^3(x^3 - y^3)^3. In this paper we give several proofs of Gleason's theorem.     

Good self-dual quasi-cyclic codes exist

We show that there are long binary quasi-cyclic self-dual (either Type I or Type II) codes satisfying the Gilbert-Varshamov bound.     

Shadow bounds for self-dual codes

Conway and Sloane (1990) have previously given an upper bound on the minimum distance of a singly-even self-dual binary code, using the concept of the shadow of a self-dual code. We improve their bound, finding that the minimum distance of a self-dual binary code of length n is at most 4[n/24]+4, except when n mod 24=22, when the bound is 4[n/24]+6. We also show that a code of length a multiple of 24 meeting the bound cannot be singly-even. The same technique gives similar results for additive codes over GF(4) (relevant to quantum coding theory)     

A systematic construction of self-dual codes

A new coding construction scheme of block codes using short base codes and permutations that enables the construction of binary self-dual codes is presented in Cadic et al. (2001) and Carlach et al. (1999, 2000). The scheme leads to doubly-even (resp,. singly-even) self-dual codes provided the base code is a doubly-even self-dual code and the number of permutations is even (resp., odd). We study the particular case where the base code is the [8, 4, 4] extended Hamming. In this special case, we construct a new [88, 44, 16] extremal doubly-even self-dual code and we give a new unified construction of the five [32, 16, 8] extremal doubly-even self-dual codes.     

Weight distributions of cyclic self-dual codes

We give a 1-level squaring construction for binary repeated-root cyclic codes of length n=2/sup a/b, a/spl ges/1, b odd. This allows us to obtain the weight distributions of all cyclic binary self-dual codes of lengths up to 110, which are not accessible by direct computation. We also use the shadow construction, as a particular method for Type I self-dual codes.     

A note on self-dual group codes

We classify group algebras over Galois rings containing self-dual ideals; i.e., ideals C which satisfy C = C/sup /spl perp// with respect to the natural nondegenerate bilinear form given on group algebras.     

Existence of some extremal self-dual codes

Two new singly-even extremal, self-dual codes are constructed: a [52,26,10] code and a [54,27,10] code     

New extremal self-dual codes of length 62 and related extremalself-dual codes

New extremal self-dual codes of length 62 are constructed with weight enumerators of three different types. Two of these types were not represented by any known code up till now. All these codes possess an automorphism of order 15. Some of them are used to construct extremal self-dual codes of length 60 by the method of subtracting. By additional subtracting, an extremal self-dual [58, 29, 10] code was obtained having a weight enumerator which does not correspond to any code known so far     

Soft decoding performance of extremal self-dual codes

Performance of soft decoded extremal self-dual codes of lengths 8 to 72 are obtained over the Gaussian channel. The results indicate that for decoded bit-error rates below 10^-3, which is the main region of interest for coding application, substantial coding gains can be obtained by using extremal self-dual codes.     

New asymptotic bounds for self-dual codes and lattices

We give an independent proof of the Krasikov-Litsyn bound d/n/spl lsim/(1-5/sup -1/4/)/2 on doubly-even self-dual binary codes. The technique used (a refinement of the Mallows-Odlyzko-Sloane approach) extends easily to other families of self-dual codes, modular lattices, and quantum codes; in particular, we show that the Krasikov-Litsyn bound applies to singly-even binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O(/spl radic/n).     

On existence of good self-dual quasicyclic codes

For a long time, asymptotically good self-dual codes have been known to exist. Asymptotically good 2-quasicyclic codes of rate 1/2 have also been known to exist for a long time. Recently, it was proved that there are binary self-dual n/3-quasicyclic codes of length n asymptotically meeting the Gilbert-Varshamov bound. Unlike 2-quasicyclic codes, which are defined to have a cyclic group of order n/2 as a subgroup of their permutation group, the n/3-quasicyclic c codes are defined with a permutation group of fixed order of 3. So, from the decoding point of view, 2-quasicyclic c codes are preferable to n/3-quasicyclic c codes. In this correspondence, with the assumption that there are infinite primes p with respect to (w r t.) which 2 is primitive, we prove that there exist classes of self-dual 2p-quasicyclic c codes and Type II 8p-quasicyclic c codes of length respectively 2p/sup 2/ and 8p/sup 2/ which asymptotically meet the Gilbert-Varshamov bound. When compared with the order of the defining permutation groups, these classes of codes lie between the 2-quasicyclic c codes and the n/3-quasicyclic c codes of length n, considered in previous works.