Quantum codes from concatenated algebraic-geometric codes

We apply Steane's enlargement of the Calderbank-Shor-Steane (CSS) codes and additive codes over F/sub 4/ to concatenated algebraic-geometric codes to construct many good quantum codes with fewer restrictions on the parameters compared to some known quantum codes. Some of the quantum codes we have constructed are either optimal or have parameters as good as the best known codes, while some have parameters better than those obtained from other known constructions.     

Weighted Reed-Muller codes and algebraic-geometric codes

A generalization of the Reed-Muller codes, the weighted Reed-Muller codes, is presented. The code parameters are estimated and the duals are shown also to be weighted Reed-Muller codes. It is shown how the minimum distance of certain algebraic-geometric codes in many cases can be determined exactly or an upper bound can be found, using subcodes which are weighted Reed-Muller codes     

Error-locator ideals for algebraic-geometric codes

The error locations for an algebraic-geometric code C*(D,mP) are exactly the common zeros (that is, a projective variety V(I)) of a set (ideal) I of error-locator functions. The paper gives a one-dimensional Berlekamp-Massey version of the Feng-Rao (1993) algorithm for decoding algebraic-geometric codes C*(D,mP). This produces a generating set for I (as an ideal) of size at most ρ (the smallest positive pole order at P of any function in L(mP)) relative to any error of weight at most e<½δ_m*, with δ_m*:=m-2g+2 the designed minimum distance of the code. This algorithm requires at most c(ρm²+Nρm+ρ²m) field multiplications, with c a small constant, and N a small constant function of the curve. The error-positions are then given as exactly the common zeros of generator functions of the error-locator ideal I     

Efficient Forney functions for decoding AG codes

Using a Forney formula to solve for the error magnitudes in decoding algebraic-geometric (AG) codes requires producing functions σ_P, which are 0 at all but one point P of the variety of the error-locator ideal. The best such function is produced here in a reasonably efficient way from a lex Grobner basis. This lex basis is, in turn, produced efficiently from a weighted grevlex basis by using the FGLM algorithm. These two steps essentially complete the efficient decoding scheme based on a Forney formula started in the author's previous work (see ibid., vol.42, p.1263-8, 1996)     

List decoding of algebraic-geometric codes

We generalize Sudan's (see J. Compl., vol.13, p.180-93, 1997) results for Reed-Solomon codes to the class of algebraic-geometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error-correction bound (d-1)/2, d being the minimum distance of the code. Our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields. For the latter problem we design a polynomial-time algorithm and show that the resulting overall list-decoding algorithm runs in polynomial time under some mild conditions. Several examples are included     

Which linear codes are algebraic-geometric?

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If conditions are imposed on the degree of the divisor use, then criteria are derived for linear codes to be algebraic-geometric. In particular. the family of q-ary Hamming codes is investigated, and it is proven that only those with redundancy one or two and the binary (7,4,3) code are algebraic-geometric in this sense. For these codes. the authors explicitly give a curve, rational points, and a divisor. It is proven that this triple is in a certain sense unique in the case of the (7,4,3) code.<>     

Explicit constructions of algebraic-geometric codes

We propose a simple construction of algebraic-geometric codes which are subcodes of Goppa codes and which coincide with Goppa codes in many cases. The codes we construct have the advantage that for an explicitly given extension of the rational function field, one can easily obtain explicit bases and therefore an exact formula for the dimension. Furthermore, we show that in many cases good upper and lower bounds for the minimum distance can be obtained     

On the decoding of algebraic-geometric codes

A decoding algorithm for algebraic-geometric codes arising from arbitrary algebraic curves is presented. This algorithm corrects any number of errors up to [(d-g-1)/2], where d is the designed distance of the code and g is the genus of the curve. The complexity of decoding equals σ(n³) where n is the length of the code. Also presented is a modification of this algorithm, which in the case of elliptic and hyperelliptic curves is able to correct [(d-1)/2] errors. It is shown that for some codes based on plane curves the modified decoding algorithm corrects approximately d/2-g/4 errors. Asymptotically good q-ary codes with a polynomial construction and a polynomial decoding algorithm (for q⩾361 on some segment their parameters are better than the Gilbert-Varshamov bound) are obtained. A family of asymptotically good binary codes with polynomial construction and polynomial decoding is also obtained, whose parameters are better than the Blokh-Zyablov bound on the whole interval 0<σ<1/2     

A Goppa-like bound on the trellis state complexity of algebraic-geometric codes

For a linear code C of length n and dimension k, Wolf (1978) noticed that the trellis state complexity s(C) of C is upper-bounded by w(C):=min(k,n-k). We point out some new lower bounds for s(C). In particular, if C is an algebraic-geometric code, then s(C)/spl ges/w(C)-(g-a), where g is the genus of the underlying curve and a is the abundance of the code.     

Decoding algebraic-geometric codes up to the designed minimumdistance

A simple decoding procedure for algebraic-geometric codes C _Ω(D,G) is presented. This decoding procedure is a generalization of Peterson's decoding procedure for the BCH codes. It can be used to correct any [(d*-1)/2] or fewer errors with complexity O(n³), where d * is the designed minimum distance of the algebraic-geometric code and n is the codelength     

Efficient decoding algorithms for generalized Reed-Muller codes

Previously, a class of generalized Reed-Muller (RM) codes has been suggested for use in orthogonal frequency-division multiplexing. These codes offer error correcting capability combined with substantially reduced peak-to mean power ratios. A number of approaches to decoding these codes have already been developed. Here, we present low complexity, suboptimal alternatives which are inspired by the classical Reed decoding algorithm for binary RM codes. We simulate these new algorithms along with the existing decoding algorithms using additive white Gaussian noise and two-path fading models for a particular choice of code. The simulations show that one of our new algorithms outperforms all existing suboptimal algorithms and offers performance that is within 0.5 dB of maximum-likelihood decoding, yet has complexity comparable to or lower than existing decoding approaches     

A method for decoding of generalized Goppa codes (Corresp.)

It is shown how the theory of continued fractions for polynomials can be used for the decoding of a class of codes which contains Goppa codes.     

LDPC codes from generalized polygons

We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.     

Soft-decision decoding of Reed-Muller codes as generalized multipleconcatenated codes

Constructs Reed-Muller codes by generalized multiple concatenation of binary block codes of length 2. As a consequence of this construction, a new decoding procedure is derived that uses soft-decision information. The algorithm is designed for low decoding complexity and is applicable to all Reed-Muller codes. It gives better decoding performance than soft-decision bounded-distance decoding. Its decoding complexity is much lower than that of maximum-likelihood trellis decoding of Reed-Muller codes, especially for long codes     

On generalized low-density parity-check codes based on Hammingcomponent codes

In this paper we investigate a generalization of Gallager's (1963) low-density (LD) parity-check codes, where as component codes single error correcting Hamming codes are used instead of single error detecting parity-check codes. It is proved that there exist such generalized low-density (GLD) codes for which the minimum distance is growing linearly with the block length, and a lower bound of the minimum distance is given. We also study iterative decoding of GLD codes for the communication over an additive white Gaussian noise channel. The performance in terms of the bit error rate, obtained by computer simulations, is presented for GLD codes of different lengths     

Long codes for generalized FH-OFDMA through unknown multipathchannels

A generalized frequency-hopping (GFH) orthogonal frequency-division multiple-access (OFDMA) system is developed in this paper as a structured long code direct-sequence code-division multiple-access (DS-CDMA) system in order to bridge frequency-hopped multicarrier transmissions with long code DS-CDMA. Through judicious code design, multiuser interference is eliminated deterministically in the presence of unknown frequency-selective multipath channels. Thanks to frequency-hopping, no single user suffers from consistent fading effects and constellation-irrespective channel identifiability is guaranteed regardless of channel nulls. A host of blind channel estimation algorithms are developed trading off complexity with performance. Two important variants, corresponding to slow- and fast-hopping, are also addressed with the latter offering symbol recovery guarantees. Performance analysis and simulation results illustrate the merits of GFH-OFDMA relative to conventional OFDMA and long code DS-CDMA with pseudorandom noise codes and RAKE reception     

Maximum transition run codes for generalized partial responsechannels

A new twins constraint for maximum transition run (MTR) codes is introduced to eliminate quasi-catastrophic error propagation in sequence detectors for generalized partial response channels with spectral nulls both at dc and at the Nyquist frequency. Two variants of the twins constraint that depend on whether the generalized partial response detector trellis is unconstrained or j-constrained are studied. Deterministic finite-state transition diagrams that present the twins constraint are specified, and the capacity of the new class of MTR constraints is computed. The connection between (G,I) constraints and MTR(j) constraints is clarified. Code design methodologies that are based on look-ahead coding in combination with violation detection/substitution as well as on state splitting are used to obtain several specific constructions of high-rate MTR codes     

Determination of error values for algebraic-geometry codes and theForney formula

In the decoding of one-point algebraic-geometry codes with one defining equation in two variables we generalize the Forney formula for the Bose-Chaudhuri-Hocqungham (BCH) codes     

Blind and semi-blind equalization for generalized space-time block codes

This paper presents a general framework for space-time codes (STCs) that encompasses a number of previously proposed STC schemes as special cases. The STCs considered are block codes that employ arbitrary redundant linear precoding of a given data sequence together with embedded training symbols, if any. The redundancy introduced by the linear precoding imposes structure on the received data that under certain conditions can be exploited for blind or semi-blind estimation of the transmitted sequence, a linear receiver that recovers the sequence, or both simultaneously. Algorithms based on this observation are developed for the single-user flat-fading case and then extended to handle multiple users, frequency-selective fading, as well as situations where the channel is rank deficient, or there are fewer receive than transmit antennas.     

On the equivalence of generalized concatenated codes andgeneralized error location codes

We show that the generator matrix of a generalized concatenated code (GCC code) of order L consists of L submatrices, where the lth submatrix is the Kronecker product of the generator matrices of the lth inner code and the lth outer code. In a similar way we show that the parity-check matrix of a generalized error location code (GEL code) of order L consists of L submatrices, where the lth submatrix is the Gronecker product of the parity-check matrices of the lth inner code and the lth outer code. Then we use these defining matrices to show that for any GCC code there exists an equivalent GEL code and vice versa