The weight spectrum of two families of Reed-Muller codes

Consider a binary Reed-Muller code RM(r, m) defined on the full set of binary m-tuples and let this code be punctured to the spherical layer S(b) that includes only m-tuples of a given Hamming weight b. More generally, we can consider punctured RM codes RM(r, m, B) restricted to some set B of several spherical layers S(b), b ∊ B. In this paper we specify this construction for the biorthogonal codes RM(1, m) and the Hadamard codes H(m). It is shown that the overall weight of any code vector in a punctured code H(m, B) is determined by the weight w of its information block. More specifically, this weight depends only on the values of the Krawtchouk polynomials K b m(w) for all b ∊ B. We further refine our codes by limiting the possible weights w of the input information blocks. As a result, we obtain sequences of codes that meet or closely approach the Griesmer bound.

McEliece cryptosystem is an public-key cryptosystem; its security is based on the complexity of decoding problem for an arbitrary error-correcting code. V. M. Sidel’nikov in 1994 suggested to construct the cryptosystem on the base of binary Reed-Muller code. In 2007 L. Minder and A. Shokrollahi had designed a structural attack on theMcEliece cryptosystem based on the Reed-Muller codes.Herewe improve their attack and suggest a polynomial attack on the McEliece cryptosystem based on Reed-Muller codes RM(r, m) such that GCD(r,m − 1) = 1.

Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube F <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> . In this paper, we study punctured Reed-Muller codes P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> (m, b) whose positions form a spherical b-layer and include all m-tuples of a given Hamming weight b. These punctured codes inherit some recursive properties of the original RM codes and can be built from the shorter codes, by decomposing a spherical b-layer into sub-layers of smaller dimensions. However, codes P <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> (m, b) cannot be formed by the recursive Plotkin construction. We analyze recursive properties of these codes and find their code distances for arbitrary values of parameters r, m, and b.

Error exponents are studied for the recursive and majority decoding algorithms of general Reed-Muller codes RM(r, m) used on the AWGN channels. Both algorithms have low decoding complexity and substantially outperform bounded distance decoding in their error-correcting capabilities. We obtain asymptotically tight upper bounds on the output error rate that hold for both algorithms and can be used for any RM-code.

Linear tail-biting trellises for block codes are considered. By introducing the notions of subtrellis, merging interval, and sub-tail-biting trellis, some structural properties of linear tail-biting trellises are proved. It is shown that a linear tail-biting trellis always has a certain simple structure, the parallel-merged-cosets structure. A necessary condition required from a linear code in order to have a linear tail-biting trellis representation that achieves the square root bound is presented. Finally, the above condition is used to show that for r/spl ges/2 and m/spl ges/4r-1 or r/spl ges/4 and r+3/spl les/m/spl les/[(4r+5)/3] the Reed-Muller code RM(r, m) under any bit order cannot be represented by a linear tail-biting trellis whose state complexity is half of that of the minimal (conventional) trellis for the code under the standard bit order.

Generalized multilevel constructions for binary RM(r,m) codes using projections onto GF(2 q ) are presented. These constructions exploit component codes over GF(2), GF(4),..., GF(2 q ) that are based on shorter Reed-Muller codes and set partitioning using partition chains of length-2 l codes. Using these constructions we derive multilevel constructions for the Barnes-Wall ?(r,m) family of lattices which also use component codes over GF(2), GF(4),..., GF(2 q ) and set partitioning based on partition chains of length-2 l lattices. These constructions of Reed-Muller codes and Barnes-Wall lattices are readily applicable for their efficient decoding.

On the non-minimal codewords in binary Reed–Muller codes

We study the McEliece cryptosystem with u -fold use of binary Reed–Muller codes RM(r, m) . This modification of the McEliece cryptosystem was proposed by V. M. Sidelnikov in 1994 and combines high cryptographic security, transmission rate close to one, and moderate complexity of both enciphering and deciphering. For arbitrary values of the parameters u , r , and m we give an upper bound for the cardinality of the set of public keys of this cryptosystem and calculate its exact value in the case of u = 2 and r = 1.

For any pair of integers r and m, 0 ? r ? m, we construct a class of quaternary linear codes whose binary images under the Gray map are codes with the parameters of the classical rth-order Reed-Muller code RM(r, m).

In this paper, we provide a general form for sparse generator matrices of several families of Quasi-Cyclic Low-Density Parity-Check codes. Codes of this kind have a prominent role in literature and applications due to their ability to achieve excellent performance with limited complexity. While some properties of these codes (like the girth length in their associated Tanner graphs) are well investigated, estimating their minimum distance is still an open problem. By obtaining sparse generator matrices for several families of these codes, we prove that they are also Quasi-Cyclic Low-Density Generator Matrix codes, which is an important feature to reduce the encoding complexity, and provides a useful tool for the investigation of their minimum distance.

Locally recoverable codes were introduced by Gopalan <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2012, and in the same year Prakash <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> introduced the concept of codes with locality, which are a type of locally recoverable codes. In this work we introduce a new family of codes with locality, which are subcodes of a certain family of evaluation codes. We determine the dimension of these codes, and also bounds for the minimum distance. We present the true values of the minimum distance in special cases, and also show that some elements of this family are “optimal codes”, as defined by Prakash <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i>

Four infinite families of ternary cyclic codes with a square-root-like lower bound

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.

Five infinite families of binary cyclic codes and their related codes with good parameters

Several families of binary cyclic codes with good parameters