Algebraic Decoding Algorithm Research Articles

Proximity Gaps for Reed–Solomon Codes

A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the property. In particular, no set in the collection has roughly half of its members δ-close to the property and the others δ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed–Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any δ smaller than the Johnson/Guruswami–Sudan list-decoding bound of the RS code. We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for δ smaller than the unique decoding radius. Concretely, if δ is smaller than half the minimal distance of an RS code V ⊂ 𝔽 q n , then every affine space is either entirely δ-close to the code or, alternatively, at most an ( n/q )-fraction of it is δ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed–Solomon codes (due to Berlekamp–Welch and Guruswami–Sudan) on a formal element of an affine space. This involves working with Reed–Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests.

Algebraic Decoding of Quasi-Reversible BCH Codes Using Band Matrices

A Bose-Chaudhuri-Hocquenghem (BCH) is called quasi-reversible if there are consecutive elements a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,⋯,a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> in the defining set, where a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> is a negative integer and a <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> is a positive integer. A matrix in band form presented in this article involves the use of the Newton identities and symmetric polynomial identities. This matrix is the coefficient matrix of a linear system. The solution to such a linear system is the coefficients of an error-locator polynomial for quasi-reversible BCH codes. Its computational complexity is significantly lower than BCH decoding using the linear system with coefficient matrix in column echelon form. This article also proposes a new partial syndrome matrix with radical locators and a lot of zeros. Such a matrix is still in band form. The expansion of its matrix determinant is exactly a radical-locator polynomial. Compared to the former radical-locator polynomials in the literature, the newly discovered radical-locator polynomial not only requires less storage memory but also results in computational benefits. Surprisingly, the resulting polynomials are much sparser than those for the narrow-sense BCH codes. Finally, a complete algebraic decoding algorithm for quasi-reversible BCH codes is provided. The error-locator and radical-locator polynomials of degree less than or equal to 5 are listed.

Peterson–Gorenstein–Zierler algorithm for differential convolutional codes

Differential Convolutional Codes with designed Hamming distance are defined, and an algebraic decoding algorithm, inspired by Peterson–Gorenstein–Zierler’s algorithm, is designed for them.

Coding Scheme for High-Order QAM Modulations in the Manhattan Metric

In this letter, a code construction in the Manhattan metric with a polynomial complexity algebraic decoding algorithm is proposed. It is shown that the code construction combined with a high-order QAM modulation enjoys a better code rate compared to a binary BCH code along with a competitive decoding procedure complexity. A performance gain from the use of the proposed code construction increases with the growth of an order of a QAM modulation. The code construction is mostly perspective in scenarios with high signal-to-noise ratios and decoding latency constraints, which are required for current and further generations of communications systems.

Algebraic decoding of the (41, 21, 9) quadratic residue code without determining the unknown syndromes

Quadratic residue (QR) codes have a high prospect on error correction for reliable data conveyance over channel with noise. This paper presents a fast algebraic scheme for decoding the binary (41, 21, 9) QR code for correcting up to 4 errors on an one-case-by-one-case basis, in which the calculation of unknown syndromes is eliminated and the conditions for checking for the various numbers of errors that exist in the received word are also simplified compared to those from Lin T. C. et al.’s algorithm, which is a traditional algebraic decoding algorithm (ADA). The computational complexity of the decoder performing the binary (41, 21, 9) QR code is analyzed thoroughly, which demonstrates that the proposed decoding scheme is faster, simpler, and more suitable for implementation than Lin T. C. et al.’s algorithm. Numerical emulation results demonstrate that the proposed decoding scheme achieves the same error-rate performance as Lin T. C. et al.’s algorithm, but it has a significantly decreased the complexity of this decoder in terms of the CPU time.

Generalized Subspace Subcodes With Application in Cryptology

Most codes with an algebraic decoding algorithm are derived from Reed-Solomon codes. They are obtained by taking equivalent codes, for example, generalized Reed-Solomon codes, or by using the so-called subfield subcode method, which leads to alternant codes over the underlying prime field, or over some intermediate subfield. The main advantage of these constructions is to preserve both the minimum distance and the decoding algorithm of the underlying Reed-Solomon code. In this paper, we explore in detail the subspace subcodes construction. This kind of codes was already studied in the particular case of cyclic Reed-Solomon codes. We extend this approach to any linear code over the extension of a finite field. We are interested in additive codes who are deeply connected to subfield subcodes. We characterize the duals of subspace subcodes. We introduce the notion of generalized subspace subcodes. We apply our results to generalized Reed-Solomon codes which leads to codes with interesting parameters, especially over a large alphabet. To conclude this paper, we discuss the security of the use of generalized subspace subcodes of Reed-Solomon codes in a cryptographic context.

Interpolation-Based Low-Complexity Chase Decoding Algorithms for Hermitian Codes

Algebraic-geometric (AG) codes have good error-correction capability due to their generally large code word length. However, their decoding remains complex, preventing practical applications. Addressing the challenge, this paper proposes two interpolation-based low-complexity Chase (LCC) decoding algorithms for one of the most popular AG codes-Hermitian codes. By choosing $\eta $ unreliable symbols and realizing them with the two most likely decisions, $2^\eta $ decoding test-vectors can be formulated. The first LCC algorithm performs interpolation for the common elements of the test-vectors, producing an intermediate outcome that will be shared by the uncommon element interpolation. It eliminates the redundant computation for decoding each test-vector, resulting in a low-complexity. With an interpolation multiplicity of one, the decoding is further facilitated by removing the requirement of pre-calculating the Hermitian curve’s corresponding coefficients. The second LCC algorithm is an adaptive variant of the first algorithm, where the number of test-vectors is determined by the reliability of received information. When the channel condition improves, it can reduce the complexity without compromising the decoding performance. Simulation results show that the both LCC algorithms outperform a number of existing algebraic decoding algorithms for Hermitian codes. Finally, our complexity analysis will reveal the proposals’ low-complexity feature.

An analogue of the ℤ4-Goethals code in non-primitive length

This paper constructs a cyclic ℤ4-code with a parity-check matrix similar to that of Goethals code but in length 2m + 1, for all m ≥ 4. This code is a subcode of the lifted Zetterberg code for m even. Its minimum Lee weight is shown to be at least 10, in general, and exactly 12 in lengths 33, 65. The authors give an algebraic decoding algorithm which corrects five errors in these lengths for m = 5, 6 and four errors for m > 6.

Algebraic decoding of the (71, 36, 11) quadratic residue code

In this study, a new approach is developed to facilitate faster decoding of a binary systematic (71, 36, 11) quadratic residue (QR) code. In this decoder, it simplifies the step of calculating the condition and avoids calculating the unknown syndrome, thereby yielding a fast algebraic decoder for correcting four possible errors. Moreover, while using the proposed algorithm, if uses the channel measurement information proposed by Chase to sequentially invert the bits of the received word until one of the errors is cancelled for the five-error case and apply the new algebraic decoding algorithm mentioned above to correct the remaining four errors, the algorithm has been verified through a software simulation in C-language. The simulation shows that the decoding scheme developed here is more efficient than the previous decoding algorithm developed for the (71, 36, 11) QR code and it is naturally suitable for software implementation.

Spectral Analysis of Quasi-Cyclic Product Codes

This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row and column codes. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm. More specifically, we prove an explicit (unreduced) basis of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code in terms of the generator matrix in reduced Grobner basis with respect to the position-over-term (RGB/POT) order form of the $ {\ell _{A}}$ -quasi-cyclic row code and the $ {\ell _{B}}$ -quasi-cyclic column code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where $ {\ell _{A}}= {\ell _{B}}=1$ ). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code for two special cases: i) for $ {\ell _{A}}=2$ and $ {\ell _{B}}=1$ and ii) if the row code is a one-level $ {\ell _{A}}$ -quasi-cyclic code (for arbitrary $ {\ell _{A}}$ ) and $ {\ell _{B}}=1$ . For arbitrary $ {\ell _{A}}$ and $ {\ell _{B}}$ , the Pre-RGB/POT form of the generator matrix of an $ {\ell _{A}} {\ell _{B}} $ -quasi-cyclic product code is conjectured. The spectral analysis is applied to the generator matrix of the product of an $ {\ell }$ -quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given $ {\ell }$ -quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for $ {\ell }$ -phased burst errors with guaranteed decoding radius.

On the lifted Zetterberg code

The even-weight subcode of a binary Zetterberg code is a cyclic code with generator polynomial $$g(x)=(x+1)p(x)$$g(x)=(x+1)p(x), where p(x) is the minimum polynomial over GF(2) of an element of order $$2^m+1$$2m+1 in $$GF(2^{2m})$$GF(22m) and m is even. This even binary code has parameters $$[2^m+1,2^m-2m, 6]$$[2m+1,2m-2m,6]. The quaternary code obtained by lifting the code to the alphabet $${\mathbb {Z}}_4=\{0,1,2,3\}$$Z4={0,1,2,3} is shown to have parameters $$[2^m+1,2^m-2m, d_L ]$$[2m+1,2m-2m,dL], where $$d_L \ge 8$$dL?8 denotes the minimum Lee distance. The image of the Gray map of the lifted code is a binary code with parameters $$(2^{m+1}+2,2^k,d_H)$$(2m+1+2,2k,dH), where $$d_H \ge 8$$dH?8 denotes the minimum Hamming weight and $$k=2^{m+1}-4m$$k=2m+1-4m. For $$m=6$$m=6 these parameters equal the parameters of the best known binary linear code for this length and dimension. Furthermore, a simple algebraic decoding algorithm is presented for these $${\mathbb {Z}}_4$$Z4-codes for all even m. This appears to be the first infinite family of $${\mathbb {Z}}_4$$Z4-codes of length $$n=2^m+1$$n=2m+1 with $$d_L \ge 8$$dL?8 having an algebraic decoding algorithm.

On the lifted Melas code

The binary Melas code is a cyclic code with generator polynomial g(u)=p(u)p(u)? where p(u) is a primitive polynomial of odd degree m?5 and the ? denotes reciprocation. The even-weight subcode of a Melas code has generator polynomial (u+1)g(u) and parameters [2m?1,2m?2m?2,6]. This code is lifted to ?4$\mathbb {Z}_{4}$ and the quaternary code is shown to have parameters [2m?1,2m?2m?2,dL?8], where dL denotes the minimum Lee distance. An algebraic decoding algorithm correcting all errors of Lee weight ≤3 is presented for this code. The Gray map of this quaternary code is a binary code with parameters [2m+1?2,2m+1?4m?4,dH?8] where dH is the minimum Hamming distance. For m=5,7 the minimum distance equals the minimum distance of the best known linear code for the given length and code size.

Algebraic List-Decoding of Subspace Codes

Subspace codes are collections of subspaces of a cer- tain ambient vector space over a finite field. Koetter and Kschi- schang introduced subspace codes in order to correct errors and erasures in noncoherent (random) linear network coding. They have also studied a remarkable family of subspace codes obtained by evaluating certain linearized polynomials. The Koetter–Kschi- schang subspace codes are widely regarded as the counterpart of Reed–Solomon <?Pub _interword-space=""?> codes in the domain of network error-correction. Koetter and Kschischang have furthermore devised an algebraic decoding algorithm for these codes, analogous to the Berlekamp– Welch decoding algorithm for Reed–Solomon codes.

A Unified View on Known Algebraic Decoding Algorithms and New Decoding Concepts

Known properties of cyclic codes are used to give a unified description of many classical decoding algorithms for Reed-Solomon codes up to half the minimum distance. This description allows also simplified proofs for these decoders. Further, a novel decoding algorithm is derived using these properties directly and variants of a new error/erasure decoding algorithm are given. For decoding beyond half the minimum distance, a basis of all solutions for decoding is derived. This basis allows to use side information in order to decode beyond half the minimum distance. Other methods where this basis can be used are power decoding, also known as virtual syndrome extension, where additional equations are generated by taking powers of the received symbols, and interleaved Reed-Solomon codes. The extended Euclidean algorithm, which calculates the greatest common divisor, plays an essential role in many presented methods.

Algebraic decoding of the binary systematic (71, 36, 11) quadratic residue code

A fast and efficient algebraic decoding algorithm (ADA) is proposed to correct up to five possible error patterns in the binary systematic (71, 36, 11) quadratic residue (QR) code. The technique required here is based on the ADA developed by He et al. [2001. Decoding the (47, 24, 11) quadratic residue code. IEEE transactions on information theory, 47 (3), 1181–1186.] and the modification of the ADA developed by Lin et al. [2010. Decoding of the (31, 16, 7) quadratic residue code. Journal of the Chinese institute of engineers, 33 (4), 573–580; 2010. High speed decoding of the binary (47, 24, 11) quadratic residue code. Information sciences, 180 (20), 4060–4068]. The new proposed conditions and the error-locator polynomials for different numbers of errors in the received word are derived. Simulation results show that the decoding speed of the proposed ADA is faster than the other existing ADAs.

Grobner Bases for Lattices and an Algebraic Decoding Algorithm

In this paper we present Grobner bases for lattices given in a general form, including integer and non-integer lattices. Grdot{o}bner bases for binary linear codes were introduced by Borges-Quintana et al. . We extend their work to non-binary group block codes. Then, given a lattice Λ and its associated label code L, which is a group code, we define an ideal for L. A Grobner basis is assigned to Λ as the Grobner basis of its label code L. Since the associated label code for integer and non-integer lattices are group codes, the assigned Grobner bases can be obtained for both cases. Using this Grobner basis an algebraic decoding algorithm is introduced. We provide an example of the decoding method for a lower dimension lattice. We explain that the complexity of this decoding method depends on the division algorithm and show this decoding method has polynomial time complexity. Experiments for some versions of root lattices (E_7 and E_8) show that for low SNR the performance of these lattices is near to the lower bounds given in .

On Decoding of the (89, 45, 17) Quadratic Residue Code

In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> .

Algebraic decoding of negacyclic codes over $${\mathbb Z_4}$$

In this article we investigate Berlekamp's negacyclic codes and discover that these codes, when considered over the integers modulo 4, do not suffer any of the restrictions on the minimum distance observed in Berlekamp's original papers: our codes have minimum Lee distance at least 2t + 1, where the generator polynomial of the code has roots ?, ? 3, . . . , ? 2t-1 for a primitive 2nth root ? of unity in a Galois extension of $${\mathbb {Z}_4}$$ ; no restriction on t is imposed. We present an algebraic decoding algorithm for this class of codes that corrects any error pattern of Lee weight ? t. Our treatment uses Grobner bases, the decoding complexity is quadratic in t.

Algebraic decoding of the (73, 37, 13) quadratic residue code

In this study, an efficient and fast algebraic decoding algorithm (ADA) for the binary systematic quadratic residue (QR) code of length 73 with the reducible generator polynomial to correct up to six errors is proposed. The S(I, J) matrix method given by He et al. (2001) is utilised to compute the unknown syndromes S5. A technique called swap base is proposed to correct the weight-4 error patterns. To correct the weight-5 error patterns, the new error-locator polynomials for decoding the five errors are derived. Finally, the modified shift-search algorithm (SSA) developed by Lin et al. (2010) is applied to correct the weight-6 error patterns. Moreover, the computations of all syndromes are achieved in a small finite field. Simulation results show that the proposed ADA is practical.

The Scientific Legacy of Ralf Koetter

This special issue is intended as a tribute to the work of Ralf Koetter. Ralf's premature death on February 2, 2009 deprived our community not only of one of its most beloved members, but also of an imaginative individual contributor, valued collaborator, and respected leader who had already had a significant impact on coding theory and practice, and who could have potentially contributed so much more. The contributions of Ralf Koetter to our field have been remarkably diverse and often seminal. They span a wide range of areas, from algebraic decoding algorithms, to codes on graphs, to signal processing, to network coding. This editorial provides an overviewof the most salient of his results. The overview is organized into four sections, presented below roughly in the order in which Ralf's interests developed during his career: Algebraic coding theory; Codes on graphs and iterative coding; Communications and signal processing; and Network coding.