Negacyclic codes of length 2/sup s/ over galois rings

Codes over the ring of integers modulo 4 have been studied by many researchers. Negacyclic codes such that the length n of the code is odd have been characterized over the alphabet Zopf₄, and furthermore, have been generalized to the case of the alphabet being a finite commutative chain ring. In this paper, we investigate negacyclic codes of length 2^s over Galois rings. The structure of negacyclic codes of length 2^s over the Galois rings GR(2^a,m), as well as that of their duals, are completely obtained. The Hamming distances of negacyclic codes over GR(2^a,m) in general, and over Zopf₂ ^a in particular are studied. Among other more general results, the Hamming distances of all negacyclic codes over Zopf₂ ^a of length 4,8, and 16 are given. The weight distributions of such negacyclic codes are also discussed     

Consta-Abelian codes over Galois rings

We study n-length consta-Abelian codes (a generalization of the well-known Abelian codes and constacyclic codes) over Galois rings of characteristic p/sup a/, where n and p are coprime. A twisted discrete Fourier transform (DFT) is used to generalize transform domain results of Abelian and constacyclic codes, to consta-Abelian codes. Further, we characterize consta-Abelian codes invariant under two kinds of monomials, whose underlying permutations are effected by: i) multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and ii) shifting the coordinates by t positions. All the codes studied here belong to the class of quasi-twisted codes which are known to contain some good codes. We show that the dual of a consta-Abelian code invariant under the two monomials is also a consta-Abelian code closed under both monomials.     

System-theoretic properties of convolutional codes over rings

Convolutional codes over rings are particularly suitable for representing codes over phase-modulation signals. In order to develop a complete structural analysis of this class of codes, it is necessary to study rational matrices over rings, which constitutes the generator matrices (encoders) for such convolutional codes. Noncatastrophic, minimal, systematic, and basic generator matrices are introduced and characterized by using a canonical form for polynomial matrices over rings. Finally, some classes of convolutional codes, defined according to the generator matrix they admit, are introduced and analyzed from a system-theoretic point of view     

形式幂级数环中的循环码

研究了形式幂级数环与有限链环上的循环码与负循环码,利用环同构与交换图技术得到这2类环上循环码与负循环码,以及Dougherty等得到的形式幂级数环上的循环码的投影码也是循环码的结果,给出了形式幂级数环上码为循环码的一个充要条件。借助这一条件,得到了含有形式幂级数环的中国积中循环码的投影码的循环性。     

Constacyclic and cyclic codes over finite chain rings

The problem of Gray image of constacyclic code over finite chain ring is studied. A Gray map between codes over a finite chain ring and a finite field is defined. The Gray image of a linear constacyclic code over the finite chain ring is proved to be a distance invariant quasi-cyclic code over the finite field. It is shown that every code over the finite field, which is the Gray image of a cyclic code over the finite chain ring, is equivalent to a quasi-cyclic code.     

Cyclic and negacyclic codes over finite chain rings

The structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring R are established when n is not divisible by the characteristic of the residue field R~. Some cases where n is divisible by the characteristic of the residue field R~ are also considered. Namely, the structure of negacyclic codes of length 2/sup t/ over /spl Zopf//sub 2//sup m/ and that of their duals are derived.     

Affine invariant extended cyclic codes over Galois rings

Recently, Blackford and Ray-Chaudhuri used transform domain techniques to permutation groups of cyclic codes over Galois rings. They used the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length 2/sup m/ over any subring of GR(4,m) to be affine invariant. Here, we use the same technique to find a set of necessary and sufficient conditions for extended cyclic codes of length p/sup m/ over any subring of GR(p/sup e/,m) to be affine invariant, for e=2 with arbitrary p and for p=2 with arbitrary e. These are used to find two new classes of affine invariant Bose-Chaudhuri-Hocquenghem (BCH) and generalized Reed-Muller (GRM) codes over Z/sub 2//sup e/ for arbitrary e and a class of affine invariant BCH codes over Z/sub p//sup 2/ for arbitrary prime p.     

Hamming metric decoding of alternant codes over Galois rings

The standard decoding procedure for alternant codes over fields centers on solving a key equation which relates an error locator polynomial and an error evaluator polynomial by a syndrome sequence. We extend this technique to decode alternant codes over Galois rings. We consider the module M={(a, b): as≡b mod x^r} of all solutions to the key equation where s is the syndrome polynomial and r, is the number of rows in a parity-check matrix for the code. In decoding we seek a particular solution (Σ, Ω)∈M which we prove can be found in a Grobner basis for M. We present an iterative algorithm which generates a Grobner basis modulo x^k+1 from a given basis modulo x^k. At the rth step, a Grobner basis for M is found, and the required solution recovered     

一类一步多数逻辑可译码的广义汉明重量谱

线性码的广义汉明重量谱描述了码在第二类窃密信道中传输的密码学特征。该文针对一类循环码在仿射置换群之下不变的一步多数逻辑可译码的广义汉明重量谱进行了研究,提出了该类码的重量谱的估计方法,并通过实例作了说明。     

Permutation decoding of abelian codes

A permutation decoding procedure for abelian codes is introduced by using the Groebner bases theory. This method is valid for decoding all the binary abelian codes. Some examples are given to show how powerful this method can be     

Abelian codes over Galois rings closed under certain permutations

We study n-length Abelian codes over Galois rings with characteristic p/sup a/, where n and p are relatively prime, having the additional structure of being closed under the following two permutations: (i) permutation effected by multiplying the coordinates with a unit in the appropriate mixed-radix representation of the coordinate positions and (ii) shifting the coordinates by t positions. A code is t-quasi-cyclic (t-QC) if t is an integer such that cyclic shift of a codeword by t positions gives another codeword. We call the Abelian codes closed under the first permutation as unit-invariant Abelian codes and those closed under the second as quasi-cyclic Abelian (QCA) codes. Using a generalized discrete Fourier transform (GDFT) defined over an appropriate extension of the Galois ring, we show that unit-invariant Abelian and QCA codes can be easily characterized in the transform domain. For t=1, QCA codes coincide with those that are cyclic as well as Abelian. The number of such codes for a specified size and length is obtained and we also show that the dual of an unit-invariant t-QCA code is also an unit-invariant t-QCA code. Unit-invariant Abelian (hence unit-invariant cyclic) and t-QCA codes over Galois field F/sub p//sup l/ and over the integer residue rings are obtainable as special cases.     

Construction of block modulation codes over rings for fading channels

A new methodology is described for construction of block coded modulation (BCM) over modulo-8 rings for fading channels. Code construction criteria are defined to obtain the maximum use of channel characteristics and to achieve the phase invariance property. Some code examples are presented.<>     

Multistage list decoding of generalized Reed-Solomon codes over Galois rings

A two-stage list decoder for generalized Reed-Solomon codes over GR (p/sup l/,m) that can exceed the Guruswami-Sudan decoding radius t/sub GS/ with significant probability for p=2, l/spl ges/2 and m=1 was recently proposed. It makes a distinction between error values which are units and those which are zero divisors in order to exceed t/sub GS/. This letter presents an extension of that approach by exploiting the fact that each element of GR (p/sup l/,m) has a unique p-adic expansion, culminating in a multistage decoder that outperforms the two-stage decoder.     

Chinese product of constacyclic and cyclic codes over finite rings

In this paper, we study the Gray images of the Chinese product of constacyclic and cyclic codes over a finite ring. We first introduce the Chinese product of constacyclic and cyclic codes over the finite ring. We then define a Gray map between codes over the finite ring and a finite field. We prove that the Gray image of the Chinese product of constacyclic codes over the finite ring is a distance-invariant quasi-cyclic code over the finite field. We also prove that each code over the finite field, which is the Gray image of the Chinese product of cyclic codes over the finite ring, is permutation equivalent to a quasi-cyclic code.     

Interleavers for turbo codes using permutation polynomials over integer rings

A class of deterministic interleavers for turbo codes (TCs) based on permutation polynomials over /spl Zopf//sub N/ is introduced. The main characteristic of this class of interleavers is that they can be algebraically designed to fit a given component code. Moreover, since the interleaver can be generated by a few simple computations, storage of the interleaver tables can be avoided. By using the permutation polynomial-based interleavers, the design of the interleavers reduces to the selection of the coefficients of the polynomials. It is observed that the performance of the TCs using these permutation polynomial-based interleavers is usually dominated by a subset of input weight 2m error events. The minimum distance and its multiplicity (or the first few spectrum lines) of this subset are used as design criterion to select good permutation polynomials. A simple method to enumerate these error events for small m is presented. Searches for good interleavers are performed. The decoding performance of these interleavers is close to S-random interleavers for long frame sizes. For short frame sizes, the new interleavers outperform S-random interleavers.     

Serially concatenated continuous phase modulation with convolutional codes over rings

In this paper, we investigate serially concatenated continuous phase modulation (SCCPM) with convolutional codes (CC) over rings. The transmitted signals are disturbed by additive white Gaussian noise. The properties for systems with both infinite and finite block lengths are investigated. For an infinite-length system, we check the convergence threshold using the extrinsic information transfer chart. For a finite-length system, we use union-bound techniques to estimate the error floors. In the union-bound analysis, we consider both the order and the position of nonzero permuted symbols. A simple method for determining a CPM error event through the sum of the input symbol sequence is shown. Thus, we can determine if the output symbol sequence of an error event in the ring CC can form an error event in CPM. Two properties concerning the interleaver gain (IG) are investigated. A recursive search algorithm for the maximal IG is shown. Compared with previous SCCPM with a binary CC, the proposed system shows an improvement concerning the convergence threshold or error floors.     

Lee metric codes over integer residue rings (Corresp.)

A method for constructing Lee metric codes over arbitrary alphabet sizes using the elementary concepts of module theory is presented. The codes possess a high degree of symmetry. Codes with two information symbols over arbitrary alphabet sizes are cyclic reversible. For alphabet sizes which are a power of two or an odd prime number, codes with one information symbol are reversible and equidistant, and codes having more than two information symbols are quasi-cyclic reversible. Binary Reed-Muller codes arise as subcodes of the codes presented. A method of constructing equidistant Lee metric codes analogous to maximum length shift register codes is presented.     

Joint source and channel coding using turbo codes over rings

Turbo codes are a practical solution for achieving large coding gains. We present a new turbo coding scheme where the component codes are convolutional codes (CCs) over the ring of integers modulo M, with M being the alphabet size of the source encoder. The a priori knowledge of the source statistics is used during the iterative decoding procedure for improved decoder performance. As an example of application, we examine differential pulse code modulation (DPCM) encoded image transmission     

Multisequence shift register synthesis over commutative rings with identity with applications to decoding cyclic codes over integer residue rings

We present a new algorithm for solving the multisequence shift register synthesis problem over a commutative ring R with identity. Given a finite set of R-sequences, each of length L, the complexity of our algorithm in terms of R-multiplications is O(L/sup 2/) as L /spl rarr/ /spl infin/. An important application of this algorithm is in the decoding of cyclic codes over Z/sub q/ up to the Hartmann-Tzeng bound, where q is a prime power. Characterization of the set of monic characteristic polynomials of a prescribed set of multiple syndrome sequences leads to an efficient decoding procedure, which we further extend to decode cyclic codes over Z/sub m/ where m is a product of prime powers.