An improvement of the Feng–Rao bound for primary codes

Geil, Olav; Martin, Stefano

doi:10.1007/s10623-014-9983-z

Cited by 6 publications

(16 citation statements)

References 20 publications

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“…Observe that although we will always use the above "operation" to decrease the leading monomial (meaning that lm(R) ≺ lm(S) ), we may still have monomials left in the support of R(X, Y ) which are divisible by the leading monomial of D(X, Y In such cases the method often establishes many more monomials in ≺ (F ) than those divisible by the leading monomial of F . In [8] an improved Feng-Rao bound was presented which treats in addition efficiently certain families of cases where the conditions 1. and 2. are satisfied, but 3. is not. Even though the ideal and monomial ordering studied in the present section exactly satisfy conditions 1. and 2., but not 3, the improved Feng-Rao bound produces the same information as the Feng-Rao bound in this case.…”

Section: Code Words From the Klein Curvementioning

confidence: 99%

“…3.2] where they studied the corresponding dual codes. A common property of the Feng-Rao bound for primary codes and its variants are that they can be viewed [6,8] as consequences of the footprint bound [10,7] from Gröbner basis theory. To establish more accurate information for the codes under consideration it is therefore natural to try to apply the footprint bound in a more direct way, which is exactly what we do in the present paper using ingredients from Buchberger's algorithm and by considering an exhaustive number of special cases.…”

Section: Introductionmentioning

confidence: 99%

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On affine variety codes from the Klein quartic

Geil

Özbudak

2018

Cryptogr. Commun.

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We study a family of primary affine variety codes defined from the Klein quartic. The duals of these codes have previously been treated in [12, Ex. 3.2]. Among the codes that we construct almost all have parameters as good as the best known codes according to [9] and in the remaining few cases the parameters are almost as good. To establish the code parameters we apply the footprint bound [10,7] from Gröbner basis theory and for this purpose we develop a new method where we inspired by Buchbergers algorithm perform a series of symbolic computations.

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Section: Code Words From the Klein Curvementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On affine variety codes from the Klein quartic

Geil

Özbudak

2018

Cryptogr. Commun.

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“…In this case a lower bound for the distance d can be described. Indeed, d(C s U ) ≥ d(C U ) and a lower bound for the distance of the code C U can be computed following the procedure given in [13].…”

Section: Quantum Codes From Subfield-subcodes Of Affine Variety Codesmentioning

confidence: 99%

“…For simplicity, a code given by U is also called U . U 4 = {(2, 4, 0), (4, 1, 0), (1, 2, 0), (6, 0, 2), (5, 0, 1), (3, 0, 2), (6, 0, 1), (5, 0, 2), (3, 0, 1), (6, 2, 0), (5, 4, 0), (3, 1, 0), (0, 6, 2), (0, 5, 1), (0, 3, 2), (0, 6, 1), (0, 5, 2), (0, 3, 1), (2, 0, 0), (4, 0, 0), (1, 0, 0)} 2 6 1 7 7 3 4,8,16,9,18,13, 3, 6, 12, 1} 2 11 1 23 --Code / Subset p r s N 1 N 2 N 3 U 6 = {(0, 2, 0), (0, 4, 0), (0, 1, 0), (0, 6, 2), (0, 5, 1), (0, 3, 2), (0, 6, 1), (0, 5, 2), (0, 3, 1), (2, 6, 2), (4, 5, 1), (8, 3, 2), (7, 6, 1), (5, 5, 2), (1, 3, 1), (2, 1, 0), (4, 2, 0), (8, 4, 0), (7, 1, 0), (5, 2, 0), (1, 4, 0)} 2 6 1 9 7 3 U 7 = {(2, 4, 1), (4, 1, 2), (8, 2, 1), (7, 4, 2), (5, 1, 1), (1, 2, 2), (0, 6, 2), (0, 5, 1), (0, 3, 2), (0, 6, 1), (0, 5, 2), (0, 3, 1), (2, 6, 2), (4, 5, 1), (8, 3, 2), (7, 6, 1), (5, 5, 2), (1, 3, 1), (2, 2, 0), (4, 4, 0), (8, 1, 0), (7, 2, 0), (5, 4, 0), (1, 1, 0)} 2 6 1 9 7 3 U 8 = {(2, 3, 0), (4, 6, 0), (8, 5, 0), (7, 3, 0), (5, 6, 0), (1, 5, 0), (6, 6, 0), (3, 5, 0), (6, 3, 0), (3, 6, 0), (6, 5, 0), (3, 3, 0), (2, 3, 1), (4, 6, 2), (8, 5, 1), (7, 3, 2), (5, 6, 1), (1, 5, 2), (2, 4, 2), (4, 1, 1), (8, 2, 2), (7, 4, 1), (5, 1, 2), (1, 2, 1), (6, 2, 2), (3, 4, 1), (6, 1, 2), (3, 2, 1), (6, 4, 2), (3, 1, 1)} 2 6 1 9 7 3 U 9 = {(0, 2), (0, 4), (0, 1), (14, 0), (28, 0), (25, 0), (19, 0), (7, 0), (22,6), (13,5), (26,3), (21,6), (11,5), (22,3), (13,6), (26,5), (21,3), (11,6), (22,5), (13,3), (26,6), (21,5) U 12 = {(0, 6, 2), (0, 5, 4), (0, 3, 1), (6, 2, 2), (3,…”

Section: Examplesmentioning

confidence: 99%

Quantum codes from affine variety codes and their subfield-subcodes

Galindo

Hernando

2014

Des. Codes Cryptogr.

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“…From (15) we can obtain a manageable bound on the RGHWs of one-point algebraic geometric codes as we now explain. This bound can even be used when one does not know H * (Q).…”

Section: One-point Algebraic Geometric Codesmentioning

confidence: 99%

Relative generalized Hamming weights of one-point algebraic geometric codes

Geil

Martin

Matsumoto

et al. 2014

2014 IEEE Information Theory Workshop (ITW 2014)

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Security of linear ramp secret sharing schemes can be characterized by the relative generalized Hamming weights of the involved codes [30,28]. In this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one-point algebraic geometric codes. As it is demonstrated, for Hermitian codes our bound is often tight. Furthermore, for these codes the relative generalized Hamming weights are often much larger than the corresponding generalized Hamming weights. * The result in this paper is in part submitted for possible presentation in IEEE Information Theory Workshop (ITW 2014) [16]. † olav@math.aau.dk ‡ stefano@math.aau.dk § ryutaroh@rmatsumoto.org ¶ diego@math.aau.dk yuanluo@sjtu.edu.cn 1 Keywords: linear code, Feng-Rao bound, Hermitian code, one-point algebraic geometric code, relative dimension/length profile, relative generalized Hamming weight, secret sharing, wiretap channel of type II.and RGHW/RDLP was only recently reported. In particular, few classes of linear codes have been examined for their RGHW/RDLP. In this paper we study RGHW of general linear codes by the Feng-Rao approach [17], and explore its consequences for one-point algebraic geometry (AG) codes [43,22] and in particular the Hermitian codes [42,40,47].The present paper starts with a discussion of known results regarding linear ramp secret sharing schemes and it continues with demonstrating that the RGHWs can also be used to express the best case information leakage. The main result of the paper is a method to estimate RGHW of one-point algebraic geometric codes. This is done by carefully applying the Feng-Rao bounds [17] for primary [1] as well as dual [11,12,37,22,32,21] codes. From this we derive a relatively simple bound which uses information on the corresponding Weierstrass semigroup [24,8]. As shall be demonstrated for Hermitian codes the new bound is often sharp. Moreover, for the same codes the RGHW are often much larger than the corresponding generalized Hamming weights (GHW) [44] which means that studies of RGHW cannot be substituted by those of GHW.The paper is organized as follows. Section 2 describes the use of RGHW in connection with linear ramp secret sharing schemes, and in connection with communication over the wiretap channel of type II. In Section 3 we apply the theory to the special case of MDS codes. In Section 4 we show -at the level of general linear codes -how to employ the Feng-Rao bounds to estimate RGHW. This method is then applied to one-point algebraic geometric codes in Section 5. We investigate Hermitian codes in Section 6, and treat the corresponding ramp secret sharing schemes in Section 7.2 Ramp secret sharing schemes and wiretap channels of type II Ramp secret sharing schemes were introduced in [4,46]. Let F q be the finite field with q elements. A ramp secret sharing scheme with t-privacy and rreconstruction is an algorithm that, given an input s ∈ F ℓ q , outputs a vector x ∈ F n q , the vector of shares that we want to share among n players, such tha...

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An improvement of the Feng–Rao bound for primary codes

Cited by 6 publications

References 20 publications

On affine variety codes from the Klein quartic

On affine variety codes from the Klein quartic

Quantum codes from affine variety codes and their subfield-subcodes

Relative generalized Hamming weights of one-point algebraic geometric codes

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