Consider a binary Reed-Muller code defined on the hypercube and let all code positions be restricted to the -tuples of a given Hamming weight . In this paper, we specify this single-layer construction obtained from the biorthogonal codes and the Hadamard codes . 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 of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials . We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code is achieved at the minimum input weight for any . We further refine our codes by limiting the possible weights 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.
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 m b (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.
Consider a binary Reed-Muller code RM(r, m) defined on the m-dimensional hypercube F m 2 . In this paper, we study punctured Reed-Muller codes Pr(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 Pr(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.
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