Minimum distance of Hermitian two-point codes

Park, Seungkook

doi:10.1007/s10623-009-9361-4

Cited by 23 publications

(23 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that also for two-points Hermitian codes there are geometrical descriptions regarding minimum weight codewords. In particular, in 2010, using a method based on [19], Park [29] finds the minimum distance of two-points Hermitian codes and obtains a geometrical characterization of minimum weight codewords as multiplications of conics and lines. Three years later Ballico and Ravagnani [2] using different techniques describe the minimum-weight codewords of some two-points Hermitian codes through an explicit geometric characterization of their supports.…”

Section: Comparison With the Known Resultsmentioning

confidence: 99%

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Marcolla

Roggero

2019

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

Let H be the Hermitian curve defined over a finite field F q 2 . In this paper we complete the geometrical characterization of the supports of the minimumweight codewords of the algebraic-geometry codes over H, started in [26]: if d is the distance of the code, the supports are all the sets of d distinct F q 2 -points on H complete intersection of two curves defined by polynomials with prescribed initial monomials w.r.t. DegRevLex.For most Hermitian codes, and especially for all those with distance d ≥ q 2 − q studied in [26], one of the two curves is always the Hermitian curve H itself, while if d < q the supports are complete intersection of two curves none of which can be H.Finally, for some special codes among those with intermediate distance between q and q 2 − q, both possibilities occur.We provide simple and explicit numerical criteria that allow to decide for each code what kind of supports its minimum-weight codewords have and to obtain a parametric description of the family (or the two families) of the supports.2010 Mathematics Subject Classification. 11G20,11T71.

show abstract

Section: Comparison With the Known Resultsmentioning

confidence: 99%

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Marcolla

Roggero

2019

Journal of Pure and Applied Algebra

View full text Add to dashboard Cite

show abstract

“…Since then, the results were extended to two-point Hermitian codes by Homma and Kim [14,15,16,17] exploiting a similar proof to [37]. After that, in 2010 Park [30], using a method based on [19], gives a short and easy proof of the same results of Homma and Kim and obtains a geometrical characterization of minimum weight codewords as multiplications of conics and lines. The description of minimum-weight codewords of some two-points Hermitian codes have been found three years later by Ballico and Ravagnani [2] using different techniques.…”

Section: Introductionmentioning

confidence: 97%

Hermitian codes and complete intersections

Marcolla

Roggero

2020

Finite Fields and Their Applications

View full text Add to dashboard Cite

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.

show abstract

“…, m r )|: m 1 < m 2 < · · · < m r ∈Λ ≤0 }. As is well known [21,22], δ 1 is the true minimum distance of the code, and δ 2 is the true second generalized Hamming weight of the code according to Homma and Kim [23]. By Munuera's arithmetic interpretation [24], we verified that δ 3 is also the true third generalized Hamming weight, which is realized by the linear space spanned by the three codewords ev(μ 1 ) = (0, 0, 0, 0, 0, α 7 , α, 2, 0, α 7 , 0, α 6 , 1, α 3 , 0, 1, 0, 0, α 2 , 0, 0, α 7 , 0, 2, α 6 , α), ev(μ 2 ) = (0, 0, 0, 0, 0, α 7 , 2, α 5 , 0, α, 0, 2, α, α 2 , 0, 1, 0, 0, 1, 0, 0, α 3 , 0, α 6 , α 3 , 2), ev(μ 3 ) = (0, 0, 0, 0, 0, α 3 , 1, α, 0, 1, 0, α 2 , α 7 , 1, 0, α, 0, 0, α 3 , 0, 0, 1, 0, 1, α 5 , α 6 ),…”

Section: A Hermitian Codementioning

confidence: 92%

Bounds for generalized Hamming weights of general AG codes

Lee

2015

Finite Fields and Their Applications

View full text Add to dashboard Cite

We present a good bound for the generalized Hamming weights of multi-point evaluation and differential AG codes. It is a natural generalization of the order bound for onepoint AG codes. As an example, we determine the third generalized Hamming weight of one particular two-point evaluation Hermitian code. Then we extend the bound for the relative generalized Hamming weights, which are essential measures of the performance of secret sharing schemes based on linear codes. The access structure of one particular secret sharing scheme based on the previous two-point Hermitian code and its subcode is analyzed using the new bound.

show abstract

Minimum distance of Hermitian two-point codes

Abstract: We prove a formula for the minimum distance of two-point codes on a Hermitian curve.

Cited by 23 publications

References 13 publications

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Minimum-weight codewords of the Hermitian codes are supported on complete intersections

Hermitian codes and complete intersections

Bounds for generalized Hamming weights of general AG codes

Contact Info

Product

Resources

About