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.     

A Unified Method for Determining the Weight Enumerators of Binary Extended Quadratic Residue Codes

This letter proposes an improved and unified method to determine the weight enumerators of binary extended quadratic residue (EQR) codes. It is faster than the previous methods for some of binary EQR codes. Moreover, all the results for the weight enumerators of binary EQR codes are listed.     

New extremal binary [44,22,8] codes

One of the problems in coding theory is constructing self-dual codes whose weight enumerators are not yet known to exist. We use the concept of neighbors and construct extremal binary self-dual [44,22,8] codes whose weight enumerators are not yet known to exist.     

Polynomial invariants of quantum codes

The weight enumerators (Shor and Laflamme 1997) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S_kⁿ. We then present a number of equations and inequalities in these invariants; in particular, we give a higher order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4, 4, 2))₂ code are uniquely determined, an important step in a proof that any ((4, 4, 2))₂ code is additive (Rains 1999)     

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     

A relation between the row weight and column weight distributions of a matrix (Corresp.)

A reformulation of MacWilliams type of relation between the split-weight enumerators of a pair of dual codes is shown to exhibit an identity on the row weight and column weight distributions of a matrix.     

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.     

The minimum distance of the [38,19] ternary extended QR-code is 11 (Corresp.)

The method of Mykkeltveit, Lam, and McEliece for finding weight enumerators of binary QR-codes is used to prove that the minimum distance of the[38,19]ternary extended QR-code is11.     

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.     

Complete joint weight enumerators and self-dual codes

We define the complete joint weight enumerator in genus g for codes over /spl Zopf//sub 2k/ and use it to study self-dual codes and their shadows. These weight enumerators are related to the theta series of the associated lattices and Siegel and Jacobi forms are formed from these series.     

Jacobi polynomials for singly even self-dual codes and the coveringradius problems

In this correspondence, we develop a method to determine the complete coset weight distributions of the class of singly even self-dual binary codes. Our basic tool is the Jacobi polynomials for the code. It describes and controls the coset weight enumerators. As the results of our present method, we give the complete coset weight distributions of some extremal singly even self-dual codes of lengths 14, 22, 32, 36, and 40, respectively. We give the generator matrices of the used codes of lengths 36 and 40, respectively     

Quantum error detection .I. Statement of the problem

This paper is devoted to the problem of error detection with quantum codes. We show that it is possible to give a consistent definition of the undetected error event. To prove this, we examine possible problem settings for quantum error detection. Our goal is to derive a functional that describes the probability of undetected error under natural physical assumptions concerning transmission with error detection with quantum codes. We discuss possible transmission protocols with stabilizer and unrestricted quantum codes. The set of results proved in the paper shows that in all the cases considered the average probability of undetected error for a given code is essentially given by one and the same function of its weight enumerators. We examine polynomial invariants of quantum codes and show that coefficients of Rains's (see ibid., vol44, p.1388-94, 1998) “unitary weight enumerators” are known for classical codes under the name of binomial moments of the distance distribution. As in the classical situation, these enumerators provide an alternative expression for the probability of undetected error     

An efficient algorithm to compute the Euclidean distance spectrum of a general intersymbol interference channel and its applications

We present an efficient algorithm to compute the distance spectrum of a general finite intersymbol interference (ISI) channel, whose complexity is lower than those of existing methods. Closed-form expressions are derived for both input-output Euclidean distance enumerators and asymptotic distance spectrum shapes for 2-tap and 3-tap ISI channels. Coded and/or precoded ISI channels are also discussed.     

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.     

New extremal self-dual codes of lengths 42 and 44

All extremal binary self-dual codes of lengths 42 and 44 which have an automorphism of order 5 with eight independent cycles are obtained up to equivalence. There are 109 inequivalent [42, 21, 8] codes with such an automorphism. All [44, 22, 8] codes that are obtained have 29 different weight enumerators     

Translates of linear codes over Z4

We give a method to compute the complete weight distribution of translates of linear codes over Z₄. The method follows known ideas that have already been used successfully by others for Hamming weight distributions. For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for the Preparata codes are both equal to ten, independent of the code length     

A classification of posets admitting the MacWilliams identity

In this paper, all poset structures that admit the MacWilliams identity are classified, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. It is proved that being a hierarchical poset is a necessary and sufficient condition for a poset to admit the MacWilliams identity. An explicit relation is also derived between the P-weight distribution of a hierarchical poset code and the P~-weight distribution of the dual code.     

环R+vR+v2R上线性码的MacWilliams恒等式

通过构造Gray映射,对环R+vR+v²R上线性码进行了研究.定义了环R+vR+v²R上线性码的Lee重量及其几类重量计数器,给出了环R+vR+v²R上线性码及其对偶码之间的各种重量分布的MacWilliams恒等式.利用这些恒等式,不用求出环R+vR+v²R上线性码的对偶码便可得到对偶码的各种重量分布.     

A note on type II convolutional codes

The result of a search for the world's second type II (doubly-even and self-dual) convolutional code is reported. A rate R=4/8, 16-state, time-invariant, convolutional code with free distance d_free=8 was found to be type II. The initial part of its weight spectrum is better than that of the Golay convolutional code (GCC). Generator matrices and path weight enumerators for some other type II convolutional codes are given. By the “wrap-around” technique tail-biting versions of (32, 18, 8) Type II block codes are constructed     

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