249 research outputs found

    The weight enumerators for certain subcodes of the second order binary Reed-Muller codes

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    In this paper we obtain formulas for the number of codewords of each weight in several classes of subcodes of the second order Reed-Muller codes. Our formulas are derived from the following results: (i) the weight enumerator of the second order RM code, as given by Berlekamp-Sloane (1970), (ii) the MacWilliams-Pless identities, (iii) a new result we present here (Theorem 1), (iv) the Carlitz-Uchiyama (1957) bound, and, (iv′) the BCH bound.The class of codes whose weight enumerators are determined includes subclasses whose weight enumerators were previously found by Kasami (1967–1969) and Berlekamp(1968a, b)

    Symmetries of weight enumerators and applications to Reed-Muller codes

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    Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To appear in Advances in Mathematics of Communication

    Weight enumerators of Reed-Muller codes from cubic curves and their duals

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    Let Fq\mathbb{F}_q be a finite field of characteristic not equal to 22 or 33. We compute the weight enumerators of some projective and affine Reed-Muller codes of order 33 over Fq\mathbb{F}_q. These weight enumerators answer enumerative questions about plane cubic curves. We apply the MacWilliams theorem to give formulas for coefficients of the weight enumerator of the duals of these codes. We see how traces of Hecke operators acting on spaces of cusp forms for SL2(Z)\operatorname{SL}_2(\mathbb{Z}) play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math., 201

    The Partition Weight Enumerator of MDS Codes and its Applications

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    A closed form formula of the partition weight enumerator of maximum distance separable (MDS) codes is derived for an arbitrary number of partitions. Using this result, some properties of MDS codes are discussed. The results are extended for the average binary image of MDS codes in finite fields of characteristic two. As an application, we study the multiuser error probability of Reed Solomon codes.Comment: This is a five page conference version of the paper which was accepted by ISIT 2005. For more information, please contact the author

    Codes and Protocols for Distilling TT, controlled-SS, and Toffoli Gates

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    We present several different codes and protocols to distill TT, controlled-SS, and Toffoli (or CCZCCZ) gates. One construction is based on codes that generalize the triorthogonal codes, allowing any of these gates to be induced at the logical level by transversal TT. We present a randomized construction of generalized triorthogonal codes obtaining an asymptotic distillation efficiency γ1\gamma\rightarrow 1. We also present a Reed-Muller based construction of these codes which obtains a worse γ\gamma but performs well at small sizes. Additionally, we present protocols based on checking the stabilizers of CCZCCZ magic states at the logical level by transversal gates applied to codes; these protocols generalize the protocols of 1703.07847. Several examples, including a Reed-Muller code for TT-to-Toffoli distillation, punctured Reed-Muller codes for TT-gate distillation, and some of the check based protocols, require a lower ratio of input gates to output gates than other known protocols at the given order of error correction for the given code size. In particular, we find a 512512 T-gate to 1010 Toffoli gate code with distance 88 as well as triorthogonal codes with parameters [[887,137,5]],[[912,112,6]],[[937,87,7]][[887,137,5]],[[912,112,6]],[[937,87,7]] with very low prefactors in front of the leading order error terms in those codes.Comment: 28 pages. (v2) fixed a part of the proof on random triorthogonal codes, added comments on Clifford circuits for Reed-Muller states (v3) minor chang
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