Minimum Weights and Weight Enumerators of $\BBZ_{4}$-Linear Quadratic Residue Codes

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In order to find a minimum weight codeword in a linear code, the Multiple Impulse Method uses the Ordered Statistics Decoder of order 3 having a complexity which increases with the code dimension. This paper presents an important improvement of this method by finding a sub code of C of small dimension containing a lowest weight codeword. In the case of Binary Extended Quadratic Residue codes, the proposed technique consists on finding a self invertible permutation ( from the projective special linear group and searching a codeword having the minimum weight in the sub code fixed by (. The proposed technique gives the exact value of the minimum distance for all binary quadratic residue codes of length less than 223 by using the Multiple Impulse Method on the sub codes in less than one second. For lengths more than 223, the obtained results prove the height capacity of the proposed technique to find the lowest weight in less time. The proposed idea is generalized for BCH codes and it has permits to find the true value of the minimum distance for some codes of lengths 1023 and 2047. The proposed methods performed very well in comparison to previously known results.

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.

A new family of binary cyclic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,(n + 1)/2)</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,(n - 1)/2)</tex> codes are introduced, which include quadratic residue (QR) codes when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> is prime. These codes are defined in terms of their idempotent generators, and they exist for all odd <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n = p_{1}^{a_{1}} p_{2}^{a_{2}} \cdots p_{r}^{a_{r}}</tex> where each <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p_{i}</tex> is a prime <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\equiv \pm 1 \pmod{8}</tex> . Dual codes are identified. The minimum odd weight of a duadic <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n,(n + 1)/2)</tex> code satisfies a square root bound. When equality holds in the sharper form of this bound, vectors of minimum weight hold a projective plane. The unique projective plane of order 8 is held by the minimum weight vectors in two inequivalent <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(73,37,9)</tex> duadic codes. All duadic codes of length less than <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">127</tex> are identified, and the minimum weights of their extensions are given. One of the duadic codes of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">113</tex> has greater minimum weight than the QR code of that length.

The 1/2-rate binary quadratic residue (QR) codes, using binary phase-shift keyed (BPSK) modulation and hard decoding, are presented as an efficient system for reliable communication. Performance results of error correction are obtained both theoretically and by means of computer calculations for a number of binary QR codes. These results are compared with the commonly used 1/2-rate convolutional codes with constraint lengths from 3 to 7 for the hard-decision case. The binary QR codes of different lengths are shown to be equivalent in error-correction performance to some 1/2-rate convolutional codes, each of which has a constraint length K that corresponds to the error-control rate d/n and the minimum distance d of the QR codes. >

In this paper, we study quadratic residue (QR) codes of prime length [Formula: see text] over the ring [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd prime numbers. We analyze some basic properties of cyclic codes of length [Formula: see text] over [Formula: see text], we define QR codes by their generating idempotents. Further, we discuss the extended QR codes. We present a considerable number of good [Formula: see text]-ary codes as Gray images of QR codes over [Formula: see text] by considering the case when [Formula: see text] is an odd prime.

A cyclic code of length n over the ring Z2m of integer of modulo 2m is a linear code with property that if the codeword (c0, c1, cn-1) in ∈ C then the cyclic shift (c1, c2, c0) in math ∈ C. Quadratic residue (abbreviated QR) codes are a particularly interesting family of cyclic codes. We define such family of codes in terms of their idem potent generators and show that these codes also have many good properties which are analogous in many respects to properties of binary QR codes. Such codes constructed are self-orthogonal. And we also discuss their minimum hamming weight.

BCH (Bose, Chaudhuri, Hocquenghem) and QDC (Quadratic Double Circulant) codes are important classes of linear codes with many application domains. For large lengths, the true value of the minimum distances of these codes remains undetermined and therefore their error correcting capabilities are still unknown. The problem of searching codewords of lowest weight in linear codes is NP-hard problem and it is equivalent to the problem of the minimum distance research. Many methods have been used to attack this hardness such as genetic algorithms, simulated annealing, Multiple Impulse Method (MIM), Chen algorithm, Canteaut-Chabaud and Zimmermann algorithms. In a previous work, we have presented the method MIM-RSC to find the minimum weights by applying the MIM method on some Random sub codes of reduced dimensions. This reduction allows an efficient local search, but the important question here is which are the sub codes containing the lowest weights and how find them?. In general, the answer of this question isn’t evident and consequently many random sub codes should be exploited to have a great chance for taking a true minimum weight codeword. In this work, we present a new way to catch lowest weights by applying the Zimmermann algorithm on these random sub codes. This proposed scheme, Zimmermann-RSC, is validated on some BCH and QDC codes of known minimum weight and it is used to determine the minimum distance for many other codes of large lengths belonging to these families. The comparison between some competitors methods with the proposed scheme Zimmermann-RSC on many large BCH and QDC codes proves the efficiency of this latest in terms of the results quality and run time.

In this paper, a method to search the set of syndromes' indices needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting index sets, one computes the unknown syndromes and thus finds the corresponding error-locator polynomial by using an inverse-free Berlekamp-Massey (BM) algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is new. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity. In fact, the algebraic hard-decision and soft-decision decoding of the (89, 45, 17) QR code outperforms that of the (73, 37, 13) QR code because the former has a larger minimal distance. However, experimental results indicate that the (73, 37, 13) QR code outperforms the (89, 45, 17) QR code with much fewer arithmetic operations when using the LP-based decoding algorithms. The pseudocodewords analysis partially explains this seemingly strange phenomenon.

Before we actually construct and analyze several specific codes in detail, we will demonstrate the existence of “good” linear codes by a simple argument first introduced by E.N. Gilbert. Suppose we have a linear code with 2i codewords of block length n and a minimum nonzero weight d which contains a coset of weight w ≥d. Then we may form an augmented code, whose 2i+1 codewords are the union of the 2i original codewords and the 2i words in the coset of the original code. The new code is easily seen to be linear and its minimum nonzero weight is still d. The augmented code has 2n−i−1 cosets, and if any of these has minimum weight w≥d, then we may again augment the augmented code without decreasing d. In fact, we may augment the code again and again, each time doubling the number of its codewords, until we obtain a code whose minimum nonzero weight is greater than the weight of any of its cosets. Since each coset of the final code then has a leader of weight < d, it follows that the number of cosets cannot exceed the number of words of weight < d and KeywordsLinear CodeBlock CodeBlock LengthSpecific CodeOriginal CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Some classes of linear codes over the ring $$\mathbb {Z}_4+v\mathbb {Z}_4$$Z4+vZ4 with $$v^2=v$$v2=v are considered. Construction of Euclidean formally self-dual codes and unimodular complex lattices from self-dual codes over $$\mathbb {Z}_4+v\mathbb {Z}_4$$Z4+vZ4 are studied. Structural properties of cyclic codes and quadratic residue codes are also considered. Finally, some good and new $$\mathbb {Z}_4$$Z4-linear codes are constructed from linear codes over $$\mathbb {Z}_4+v\mathbb {Z}_4$$Z4+vZ4.

On minimum weight codewords in QR codes

We present a cyclotomic approach to the construction of all binary duadic codes of prime lengths. We calculate the number of all binary duadic codes for a given prime length and that of all duadic codes that are not quadratic residue codes. We give necessary and sufficient conditions for p such that all binary duadic codes of length p are quadratic residue (QR) codes. We also show how to determine some weights of duadic codes with the help of cyclotomic numbers.

As a result of their applications in network coding, space-time coding, and coding for criss-cross errors, matrix codes have garnered significant attention; in various contexts, these codes have also been termed rank-metric codes, space-time codes over finite fields, and array codes. We focus on characterizing matrix codes that are both efficient (have high rate) and effective at error correction (have high minimum rank-distance). It is well known that the inherent trade-off between dimension and minimum distance for a matrix code is reversed for its dual code; specifically, if a matrix code has high dimension and low minimum distance, then its dual code will have low dimension and high minimum distance. With an aim towards finding codes with a perfectly balanced trade-off, we study self-dual matrix codes. In this work, we develop a framework based on double cosets of the matrix-equivalence maps to provide a complete classification of the equivalence classes of self-dual matrix codes, and we employ this method to enumerate the equivalence classes of these codes for small parameters.

The classification of self-dual codes has been an extremely active area in coding theory since 1972. A particularly interesting class of self-dual codes is those of Type II which have high minimum distance (called extremal or near-extremal). It is notable that this class of codes contains famous unique codes: the extended Hamming code, the extended Golay code, and the extended quadratic residue code. We examine the subcode structures of Type II codes for lengths up to 24, extremal Type II codes of length 32, and give partial results on the extended quadratic residue code. We also develop a generalization of self-dual codes to Network Coding Theory and give some results on existence of self-dual network codes with largest minimum distance for lengths up to 10. Complementary Information Set (CIS for short) codes, a class of classical codes recently developed in, have important applications to Cryptography. CIS codes contain self-dual codes as a subclass. We give a new classification result for CIS codes of length 14 and a partial result for length 16.