Low-Complexity Algebraic Soft Decoding of Hermitian Codes with Re-Encoding Transform

Hermitian codes are an attractive alternative to Reed-Solomon codes for use in frequency-hop (FH) spread-spectrum packet radio networks. For a given alphabet size, a Hermitian code has a much longer block length than a Reed-Solomon code. This and other considerations suggest that Hermitian codes may be superior for certain applications. Analytical results are developed for the evaluation of the packet error probability for FH transmissions using Hermitian coding. We find there are several situations for which Hermitian codes provide much lower packet error probabilities than can be obtained with Reed-Solomon codes. In general, as the code rate decreases or the symbol alphabet size increases, the relative performance of Hermitian codes improves with respect to Reed-Solomon codes. Performance evaluations are presented for an additive white Gaussian noise channel and for a catastrophic partial-band interference channel, and the packet error probability is evaluated for both errors-only and errors-and-erasures decoding.

We consider Reed-Solomon (RS) codes whose evaluation points belong to a subfield, and give a linear-algebraic list decoding algorithm that can correct a fraction of errors approaching the code distance, while pinning down the candidate messages to a well-structured affine space of dimension a constant factor smaller than the code dimension. By pre-coding the message polynomials into a subspace-evasive set, we get a Monte Carlo construction of a subcode of Reed-Solomon codes that can be list decoded from a fraction (1-R-e) of errors in polynomial time (for any fixed e > 0) with a list size of O(1/e). Our methods extend to algebraic-geometric (AG) codes, leading to a similar claim over constant-sized alphabets. This matches parameters of recent results based on folded variants of RS and AG codes. but our construction here gives subcodes of Reed-Solomon and AG codes themselves (albeit with restrictions on the evaluation points).Further, the underlying algebraic idea also extends nicely to Gabidulin's construction of rank-metric codes based on linearized polynomials. This gives the first construction of positive rate rank-metric codes list decodable beyond half the distance, and in fact gives codes of rate R list decodable up to the optimal (1-R-e) fraction of rank errors. A similar claim holds for the closely related subspace codes studied by Koetter and Kschischang.We introduce a new notion called subspace designs as another way to pre-code messages and prune the subspace of candidate solutions. Using these, we also get a deterministic construction of a polynomial time list decodable subcode of RS codes. By using a cascade of several subspace designs, we extend our approach to AG codes, which gives the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1-R-e fraction of errors over an alphabet of constant size (that depends only on e). The list size bound is almost a constant (governed by log* (block length)), and the code can be constructed in quasi-polynomial time.

We present a new method to obtain cyclic sub-codes of algebraic geometric codes using their automorphisms. Automorphisms of algebraic geometric codes from F.K. Schmidt curves are proposed. We present an application of this method in designing frequency hopping sequences for spread spectrum systems. Algebraic geometric codes can provide sequences longer (better randomness) than the ones from Reed-Solomon codes.

Korchmaros and Nagy [Hermitian codes from higher degree places. J Pure Appl Algebra, doi: 10. 1016/j.jpaa.2013.04.002, 2013] computed the Weierstrass gap sequence G(P) of the Hermitian function field F q 2 ( H ) at any place P of degree 3, and obtained an explicit formula of the Matthews-Michel lower bound on the minimum distance in the associated differential Hermitian code C Ω ( D,mP ) where the divisor D is, as usual, the sum of all but one 1-degree F q 2 -rational places of F q 2 ( H ) and m is a positive integer. For plenty of values of m depending on q , this provided improvements on the designed minimum distance of C Ω ( D,mP ). Further improvements from G(P) were obtained by Korchmaros and Nagy relying on algebraic geometry. Here slightly weaker improvements are obtained from G(P) with the usual function-field method depending on linear series, Riemann-Roch theorem and Weierstrass semigroups. We also survey the known results on this subject.

This paper proposes two interpolation-based algebraic Chase decoding for elliptic codes. It is introduced from the perspective of computing the Gröbner basis of the interpolation module, for which two Chase interpolation approaches are utilized. They are Kötter’s interpolation and the basis reduction (BR) interpolation. By identifying η unreliable symbols, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> decoding test-vectors are formulated, and the corresponding interpolation modules can be defined. The re-encoding further helps transform the test-vectors, facilitating the two interpolation techniques. In particular, Kötter’s interpolation is performed for the common elements of the test-vectors, producing an intermediate outcome that is shared by the decoding of all test-vectors. The desired Gröbner bases w.r.t. all test-vectors can be obtained in a binary tree growing fashion, leading to a low complexity but its decoding latency cannot be contained. In contrast, the BR interpolation first performs the common computation in basis construction which is shared by all interpolation modules, and then conducts the module basis construction and reduction for all test-vectors in parallel. It results in a significantly lower decoding latency. Finally, simulation results are also presented to demonstrate the effectiveness of the proposed Chase decoding.

The paper review the public-key cryptosystems based on the error correcting codes such as Goppa code, BCH code, RS code, rank distance code, algebraic geometric code as well as LDPC code, and made the comparative analyses of the merits and drawbacks of them. The cryptosystem based on Goppa code has high security, but can be achieved poor. The cryptosystems based on other error correcting codes have higher performance than Goppa code. But there are still some disadvantages to solve. At last, the paper produce an assumption of the Niederreiter cascade combination cryptosystem based on double public-keys under complex circumstances, which has higher performance and security than the traditional cryptosystems.

Bounds for generalized Hamming weights of general AG codes

In multi-hop wireless sensor networks (WSN), the data transmitted from sensor nodes is highly susceptible to error corruption introduced by noisy channels. It is important to use some error correction schemes in order to control the errors introduced and to reduce the number of automatic requests for retransmission. The sensor nodes are typically wireless nodes with limited storage and computational power. Thus the error control schemes should be energy efficient at sensor nodes level. Most of the reported works assume that both encoding and decoding are performed at every node. The idea of this paper is to perform encoding only at the first node and decoding will be done at the base station. We assume that the base station has enough power to run complex decoding algorithms. This paper provides a model for energy consumed and the probability of receiving a correct word for multi-hop WSN at base station when considering a number of coding schemes and hops. The emphasis is on using powerful codes with low complexity encoding. Several powerful codes like Reed-Solomon (RS), list decoded RS codes, multivariate interpolation decoded RS codes (MIDRS) and Hermitian codes are investigated for this scheme. All these codes have very simple encoding based on RS type encoders and we will show that this scheme consumes less power than reported schemes while maintaining a low probability of error.

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to compress, in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan's list decoding algorithm for Reed-Solomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n2 l) and O(n7/3 l) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here l denotes the length of the list; assuming that l is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks.To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.

Hermitian codes are a class of very long algebraic‐geometric (AG) codes constructed from Hermitian curves, which outperform Reed–Solomon codes defined over the same finite fields and with the same code rates, as recently demonstrated by the authors. However, since there are no soft‐decision decoding algorithms for AG codes in the literature, the performance of Hermitian codes is limited and their potential is yet to be realized. An alternative method for achieving more significant coding gains is presented in this paper by serially concatenating long Hermitian codes with ring‐trellis‐coded modulation codes over the ring of integers ℤ4 and evaluating their performance through simulation results on the AWGN and Rayleigh fading channels. The scheme achieves large coding gains over single Hermitian and Reed–Solomon codes with no increase in bandwidth use and a performance comparable with the well‐known capacity‐approaching codes. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Low-density parity-check (LDPC) codes constructed by a sparse parity-check matrix are of very fast encoding and decoding algorithms. Another kind of codes, which improved the well-known Gilbert-Varshamov bound, are algebraic geometry codes (Goppa geometry codes) from algebraic curves over finite fields. In the note, we analyze their characteristic of the two class of codes and show that the algebraic geometric codes are seldom LDPC codes.

The order bound for general algebraic geometric codes

In this paper, we first give a relationship between generalized algebraic geometry codes (GAG codes) and algebraic geometry codes (AG codes). More precisely, we show that a GAG code is contained (up to isomorphism) in a suitable AG code. Next we recall the concept of an N 1 N 2 -automorphism group, a subgroup of the automorphism group of a GAG code. With the use of the relation we obtained between these two classes of codes, we show that the N 1 N 2 -automorphism group is a subgroup of the automorphism group of an AG code.

The Reed-Solomon (RS) codes, proposed in 1960 by Irving Reed and Gustav Solomon as a subset of error-correcting codes, have many current applications. The most significant of which are data recovery in storage systems, including hard drives, minidiscs, CDs, DVDs, Google's GFS, BigTable, and RAID 6, as well as in communication systems such as DSL, WiMAX, DVB, ATSC, and satellite communications. Additionally, RS codes are used as Bar codes in management and advertising systems, such as PDF-417, MaxiCode, Datamatrix, QR Code, and Aztec Code. Nowadays, RS codes over Galois Fields GF(2m) with base 2 are commonly used in these applications, with the GF(28) field being the most widely used. This allows all 256 values of a byte to be represented as a polynomial with 8 binary coefficients over GF(28). Considering RS codes as cyclic codes in GF(2m) fields, as well as the validity of mathematical dependencies in arbitrary field GF(pm), is a motivation to verify and generalize the idea of generating RS codes in a field with base other prime than 2. As a result, the paper derives the specific features of the construction of Reed-Solomon codes by considering them as a family of codes over any field GF(pm) whose base is a prime p other than 2. The paper also discusses the unique properties of basic arithmetic operations in the arbitrary field GF(pm), which arise from the non-uniqueness of the inverse elements a and -a in a field with base other than 2.

We propose a Reed-Solomon (RS) code construction of the 1-out-n (ring) signature scheme. It is obtained from the observation of the equivalency between the erasure correction technique of the RS code and the polynomial interpolation. The structure is very simple and yields a ring equation that can appropriately denoted by Z 1 + ... + Z n =v, which represents the summation of n evaluations of a polynomial. We also show how to extend the generic RST scheme [6] to a t-out-n threshold ring signature scheme.