6 research outputs found
The weight enumerators for certain subcodes of the second order binary Reed-Muller codes
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)
Codes and Protocols for Distilling , controlled-, and Toffoli Gates
We present several different codes and protocols to distill ,
controlled-, and Toffoli (or ) 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 . We present a randomized
construction of generalized triorthogonal codes obtaining an asymptotic
distillation efficiency . We also present a Reed-Muller
based construction of these codes which obtains a worse but performs
well at small sizes. Additionally, we present protocols based on checking the
stabilizers of 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 -to-Toffoli distillation,
punctured Reed-Muller codes for -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 T-gate to Toffoli gate code with
distance as well as triorthogonal codes with parameters
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
A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes
Boolean functions with high algebraic immunity are important cryptographic
primitives in some stream ciphers. In this paper, two methodologies for
constructing binary minimal codes from sets, Boolean functions and vectorial
Boolean functions with high algebraic immunity are proposed. More precisely, a
general construction of new minimal codes using minimal codes contained in
Reed-Muller codes and sets without nonzero low degree annihilators is
presented. The other construction allows us to yield minimal codes from certain
subcodes of Reed-Muller codes and vectorial Boolean functions with high
algebraic immunity. Via these general constructions, infinite families of
minimal binary linear codes of dimension and length less than or equal to
are obtained. In addition, a lower bound on the minimum distance of
the proposed minimal linear codes is established. Conjectures and open problems
are also presented. The results of this paper show that Boolean functions with
high algebraic immunity have nice applications in several fields such as
symmetric cryptography, coding theory and secret sharing schemes