On the small-weight codewords of some Hermitian codes

Marcolla, Chiara; Pellegrini, Marco; Sala, Massimiliano

doi:10.1016/j.jsc.2015.03.003

Cited by 13 publications

(14 citation statements)

References 18 publications

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“…So far, few partial results are known, and the first of them appears only in 2011 ( [31]). The geometric characterization of the small-weight codewords of the Hermitian code C m for a few cases of m (mainly in the first and second phase) can be found in [3,5,6,26,31]. In particular in [31] and [26], the first author of this paper and her coauthors study Hermitian codes C m with distance d ≤ q, that is with m ≤ q 2 − 2 (first phase).…”

Section: Introductionmentioning

confidence: 99%

Hermitian codes and complete intersections

Marcolla

Roggero

2020

Finite Fields and Their Applications

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In this paper we consider the Hermitian codes defined as the dual codes of one-point evaluation codes on the Hermitian curve H over the finite field F q 2 . We focus on those with distance d ≥ q 2 − q and give a geometric description of the support of their minimum-weight codewords. We consider the unique writing µq+λ(q+1) of the distance d with µ, λ non negative integers, and µ ≤ q, and consider all the curves X of the affine plane A 2 F q 2 of degree µ + λ defined by polynomials with x µ y λ as leading monomial w.r.t. the DegRevLex term ordering (with y > x). We prove that a zero-dimensional subscheme Z of A 2 F q 2 is the support of a minimum-weight codeword of the Hermitian code with distance d if and only if it is made of d simple F q 2 -points and there is a curve X such that Z coincides with the scheme theoretic intersection H ∩ X (namely, as a cycle, Z = H · X ). Finally, exploiting this geometric characterization, we propose an algorithm to compute the number of minimum weight codewords and we present comparison tables between our algorithm and MAGMA command MinimumWords.

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Section: Introductionmentioning

confidence: 99%

Hermitian codes and complete intersections

Marcolla

Roggero

2020

Finite Fields and Their Applications

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“…Both codes have minimum distance d = 3 [12]. According to [4], the two codes have the same minimum weight words. Taking α n−5 = q 3 + q 2 − 3q − 3 and α n−4 = q 3 + q 2 − 3q − 2 and constructing the kernel matrix G H as above, we see that…”

Section: Examplementioning

confidence: 99%

“…Pick the column with the longest run of zeros on the top, which is the first column of G H (4). Since the last row of G H (4) is the only row with a nonzero entry in the first column, we will remove the last row and the first column of G H (4).…”

Section: Theoremmentioning

confidence: 99%

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Exponents of polar codes using algebraic geometric code kernels

Anderson

Matthews

2014

Des. Codes Cryptogr.

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Reed-Solomon and BCH codes were considered as kernels of polar codes by Mori and Tanaka (IEEE Information Theory Workshop, 2010, pp 1-5) and Korada et al. (IEEE Trans Inform Theory 56(12):6253-6264, 2010) to create polar codes with large exponents. Mori and Tanaka showed that Reed-Solomon codes over the finite field F q with q elements give the best possible exponent among all codes of length l ≤ q. They also stated that a Hermitian code over F 2 r with r ≥ 4, a simple algebraic geometric code, gives a larger exponent than the Reed-Solomon matrix over the same field. In this paper, we expand on these ideas by employing more general algebraic geometric (AG) codes to produce kernels of polar codes. Lower bounds on the exponents are given for kernels from general AG codes, Hermitian codes, and Suzuki codes. We demonstrate that both Hermitian and Suzuki kernels have larger exponents than Reed-Solomon codes over the same field, for q ≥ 3; however, the larger exponents are at the expense of larger kernel matrices. Comparing kernels of the same size, though over different fields, we see that Reed-Solomon kernels have larger exponents than both Hermitian and Suzuki kernels. These results indicate a tradeoff between the exponent, kernel matrix size, and field size.

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“…Even the problem of computing codewords of minimum weight can be a difficult task apart from specific cases. In [13], following the approach of [1,16], the authors compute the number of minimum weight codewords of some dual AG codes from the Hermitian curve by providing an algebraic and geometric description for codewords of a given weight belonging to any fixed affine-variety code.…”

Section: Introductionmentioning

confidence: 99%