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Spherically Punctured Biorthogonal Codes

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2 Author(s)
Dumer, I. ; Coll. of Eng., Univ. of California, Riverside, Riverside, CA, USA ; Kapralova, O.

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube BB F2m 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 Kbm(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.

Published in:

Information Theory, IEEE Transactions on  (Volume:59 ,  Issue: 9 )

Date of Publication:

Sept. 2013

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