Split codes and the Mathieu groups

The Quadratic Residue (QR) codes have a rich mathematic structure. Unfortunately, their Algebraic Decoding (AD) is not generalizable for all QR codes. In this study, an efficient hard decoding algorithm is proposed to generalize the decoding of the binary systematic Quadratic Residue (QR) codes. The proposed decoder corrects t erroneous bits or less, in the received word, based on a reduced set of permutations derived from the large automorphism group of QR codes. This set of permutations is applied to the received word to move the error positions and trap all of them in redundancy. Then, to evaluate the proposed method, we applied it to many binary QR codes of moderate code length starting with 17 until 113 with reducible and irreducible generator polynomials. The proposed decoder was validated by inserting all possible error patterns, that have t or less erroneous positions, as input of the proposed decoder and the output is always a correct codeword. The complexity study, in terms of the number of operations used, reveals that the light permutation decoding LPD algorithm significantly decreases decoding complexity without performance loss. So, it is qualified to be a good competitor to decode QR codes with lower lengths but is the best for QR codes with higher lengths.

The class of binary quadratic residue (QR) codes of length n=8m-1 contains two perfect codes. These are the (7,4,3) Hamming code and the (23,12,7) Golay code. However, it is proved in the present paper that there are no quasi-perfect QR codes of length 8m-1. Finally, this result is generalized to all binary self-dual codes of length N>72. >

<p style='text-indent:20px;'>This paper deals with the problem of determining whether a PD-set exists for a given linear code <inline-formula><tex-math id="M1">\begin{document}$ \mathcal C $\end{document}</tex-math></inline-formula> and information set <inline-formula><tex-math id="M2">\begin{document}$ I $\end{document}</tex-math></inline-formula>. A computational approach is proposed and illustrated with two exceptional codes with automorphism groups isomorphic to the sporadic simple groups <inline-formula><tex-math id="M3">\begin{document}$ \mathrm{M}_{12} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ \mathrm{M}_{22} $\end{document}</tex-math></inline-formula>, respectively. In both cases, the existence of a PD–set is proven. In general, the algorithm works well whenever the code <inline-formula><tex-math id="M5">\begin{document}$ \mathcal C $\end{document}</tex-math></inline-formula> has a very large automorphism group.

In this paper, an algebraic decoding method is proposed for the quadratic residue codes that utilize the Berlekamp-Massey algorithm. By a modification of the technique developed by He et al., one can express the unknown syndromes as functions of the known syndromes. The unknown syndromes are determined by an efficient algorithm also developed in this paper. With the appearance of unknown syndromes, one obtains the consecutive syndromes that are needed for the application of the Berlekamp-Massey algorithm. The decoding scheme, developed here, is easier to implement than the previous decoding algorithm developed for the Golay code and the (47, 24, 11) QR code. Moreover, it can be extended to decode all codes of the family of binary quadratic residue codes with irreducible generating polynomials.

Unrestricted codes with the golay parameters are unique

Tail-biting trellis representations of block codes are investigated. We develop some elementary theory, and present several intriguing examples, which we hope will stimulate further developments in this field. In particular, we construct a 16-state 12-section structurally invariant tail-biting trellis for the (24, 12, 8) binary Golay code. This tail-biting trellis representation is minimal: it simultaneously minimizes all conceivable measures of state complexity. Moreover, it compares favorably with the minimal conventional 12-section trellis for the Golay code, which has 256 states at its midpoint, or with the best quasi-cyclic representation of this code, which leads to a 64-state tail-biting trellis. Unwrapping this tail-biting trellis produces a periodically time-varying 16-state rate-1/2 "convolutional Golay code" with d=8, which has attractive performance/complexity properties. We furthermore show that the (6, 3, 4) quaternary hexacode has a minimal 8-state group tail-biting trellis, even though it has no such linear trellis over F/sub 4/. Minimal tail-biting trellises are also constructed for the (8, 4, 4) binary Hamming code, the (4, 2, 3) ternary tetracode, the (4, 2, 3) code over F/sub 4/, and the Z/sub 4/-linear (8. 4, 4) octacode.

A method for decoding of the quadratic residue (QR) code with hash tables is presented. The method can be applied in decoding the (31, 16, 7) QR code that the generator polynomial can be factored. In other words, the mapping between elements of syndrome S1 and all correctable error patterns is not one-to-one. Therefore, the decoding of the (31, 16, 7) QR code needs to stuck S1,S5, S7 known syndrome mapping an error pattern that has one-to-one nature, where a subscript 1, 5, 7 are cyclotomic cosets. Furthermore, the algorithm determines the error locations by hash tables without operating the additions and multiplications over finite field. To decode QR code result, the hash table method for the (31, 16, 7) QR code is dramatically reduced the memory size above 97%. It is very suitable for high speed in modern communication system.

We study a construction of the Mathieu group M 12 using a game reminiscent of Loyd's "15-puzzle." The elements of M 12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M 13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M 12 and M 13. We develop these results, and extend them to the double covers and automorphism groups of M 12 and M 13, using the ternary Golay code and 12 × 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

Uniformly efficient trellis decoders are known for very few codes, and no general method is known that can decide whether such a decoder exists. It is shown that this question is substantially simplified in the case of self-dual codes, when certain subcodes meet the Griesmer bound with equality. Furthermore, in many cases the result makes it possible to count the number of uniformly efficient permutations. In some cases the existence and number of uniformly efficient trellises may be deduced directly from the parameters of the code. Among the codes that meet the criterion are the [24,12,8] Golay code, for which the number of uniformly efficient permutations is derived, four of the [32,16,8] doubly even codes, and the [48,24,12] quadratic residue code, for which a lower bound on the number of uniformly efficient permutations is derived.

A new algebraic algorithm to decode the ternary Golay code

A new algebraic approach for decoding the quadratic residue (QR) codes, in particular the (41, 21, 9) QR code, is presented. The key ideas behind this decoding technique are a systematic application of the Sylvester resultant method to the Newton identities associated with the syndromes to find the error-locator polynomial, and next a method for determining error locations by solving certain quadratic, cubic, and quartic equations over GF(2/sup m/) in a new way which uses Zech's logarithms for the arithmetic. The logarithms developed for Zech's logarithms save a substantial amount of computer memory by storing only a table of Zech's logarithms. These algorithms are suitable for implementation in a programmable microprocessor or special-purpose VLSI chip. It is expected that the algebraic methods developed can apply generally to other codes such as the BCH and Reed-Solomon codes. >

In this letter, the algebraic decoding algorithm of the (89, 45, 17) binary quadratic residue (QR) code proposed by Truong et al. is modified by using the efficient determination algorithm of the primary unknown syndromes. The correctness of the proposed decoding algorithm is verified by computer simulations and the use of two corollaries. Also, simulation results show that the CPU time of this algorithm is approximately 4 times faster than that of the previously mentioned decoding algorithm at least. Therefore, such a fast decoding algorithm can now be applied to achieve efficiently the reliability-based decoding for the (89, 45, 17) QR code. Finally, the performance of its algebraic soft-decision decoder expressed in terms of the bit-error probability versus E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> is given but not available in the literature.

In the context of informatization and big data, the demands for the reliability and real-time of information transmission in wireless communication channels are increasing. Quadratic residue (QR) codes have a high prospect on error correction for reliable data conveyance over channel with noise. This paper proposes an optimized algebraic scheme for decoding (71, 36, 11) QR code for correcting up to 5 errors on one-case-by-one-case basis. This scheme is mainly to improve its algebraic decoding algorithm based on the Lin’s algorithm, only in this paper, the Lin’s algorithm is called LTC71 algorithm. The key technology is to use the algebraic structure of QR code and give a new discriminant condition by mathematical derivation, which can quickly detect whether there are four errors in the code. Simulation results show that the proposed decoding scheme achieves the same bit error-rate performance as LTC71 algorithm, and the decoding time is reduced by about 28.72% owing to the reduced complexity of the decoder when there are four errors in the code.

In this paper, an effective table lookup decoding algorithm (TLDA), called the syndrome and syndrome difference decoding algorithm (SSDDA), is developed to decode the binary systematic (41, 21, 9) quadratic residue (QR) code. It is based on the property of the weight of syndrome and the weight of syndrome difference to reduce the memory size of the lookup table. The proposed algorithm requires a lookup table, called the compact lookup table (CLT), which only consists of 231 syndromes and their corresponding error patterns. The advantage of the SSDDA over the previous TLDAs is that the memory size of the proposed CLT is only about 82.2% and 2% of the lookup table needed in the decoding algorithms of Lin et al. (2010) and Chen et al. (2010), respectively.

In this paper presents an efficient table lookup algorithm for high-throughput decoding of the (71, 36, 11) Quadratic Residue (QR) code. The main ideas behind this decoding technique are based on one-to-one mapping between the syndromes S1 and correctable error patterns. As compared with the binary lookup table method, the presented technique is faster than binary searching method for finding error pattern. In addition, it has the advantage that only the operations of addition over a Finite Field are used for decoding QR codes. Moreover, the presented high speed of the decoding procedure has potential applications in modern communication systems and digital signal processing (DSP) systems.