Some results on cyclic codes which are invariant under the affine group and their applications

Abstract. We investigate the p-adic valuation of Weil sums of the form W F,d (a) = x∈F ψ(x d − ax), where F is a finite field of characteristic p, ψ is the canonical additive character of F , the exponent d is relatively prime to |F × |, and a is an element of F . Such sums often arise in arithmetical calculations and also have applications in information theory. For each F and d one would like to know V F,d , the minimum p-adic valuation of W F,d (a) as a runs through the elements of F . We exclude exponents d that are congruent to a power of p modulo |F × | (degenerate d), which yield trivial Weil sums. We prove that V F,d ≤ (2/3)[F : Fp] for any F and any nondegenerate d, and prove that this bound is actually reached in infinitely many fields F . We also prove some stronger bounds that apply when [F : Fp] is a power of 2 or when d is not congruent to 1 modulo p−1, and show that each of these bounds is reached for infinitely many F .

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“…If p = 2, and d = 2 n/ℓ + 1 or d = 2 2n/ℓ − 2 n/ℓ + 1, then it is known (see [17], [23,Theorem 5], [24,Remark 3], [25,Theorem 16], [44]…”

Section: Proof Of the Universal Upper Boundmentioning

confidence: 99%

The p-adic valuations of Weil sums of binomials

Katz

Langevin

Lee

et al. 2017

Journal of Number Theory

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“…[Kasami et al, 1967] and references therein). In the context of locally testable and locally correctable codes, affine-invariance facilitates natural local correctiong/testing procedures under minimal conditions.…”

Section: Locally Correctable and Locally Testable Codes Affine-invarmentioning

confidence: 99%

“…and e i ≤ d i for all i. The set {e|e ≤ p d} is called the p-shadow of d. The following Lemma appears as Theorem 1 in Kasami et al [1967]. We provide a proof of it, for the sake of completeness, in Section 5.…”

Section: Degree Sets Of Affine-invariant Familiesmentioning

confidence: 99%

Limits on the Rate of Locally Testable Affine-Invariant Codes

Ben–Sasson¹,

Sudan²

2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Despite its many applications, to program checking, probabilistically checkable proofs, locally testable and locally decodable codes, and cryptography, "algebraic property testing" is not wellunderstood. A significant obstacle to a better understanding, was a lack of a concrete definition that abstracted known testable algebraic properties and reflected their testability. This obstacle was removed by [Kaufman and Sudan, STOC 2008] who considered (linear) "affine-invariant properties", i.e., properties that are closed under summation, and under affine transformations of the domain. Kaufman and Sudan showed that these two features (linearity of the property and its affine-invariance) play a central role in the testability of many known algebraic properties. However their work does not give a complete characterization of the testability of affine-invariant properties, and several technical obstacles need to be overcome to obtain such a characterization. Indeed, their work left open the tantalizing possibility that locally testable codes of rate dramatically better than that of the family of Reed-Muller codes (the most popular form of locally testable codes, which also happen to be affine-invariant) could be found by systematically exploring the space of affine-invariant properties.In this work we rule out this possibility and show that general (linear) affine-invariant properties are contained in Reed-Muller codes that are testable with a slightly larger query complexity. A central impediment to proving such results was the limited understanding of the structural restrictions on affine-invariant properties imposed by the existence of local tests. We manage to overcome this limitation and present a clean restriction satisfied by affine-invariant properties that exhibit local tests. We do so by relating the problem to that of studying the set of solutions of a certain nice class of (uniform, homogenous, diagonal) systems of multivariate polynomial equations. Our main technical result completely characterizes (combinatorially) the set of zeroes, or algebraic set, of such systems.

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“…We recall a characterization of Kasami, Lin and Peterson of the extended cyclic codes which are affine-invariant in terms of the roots of the cyclic code [KLP1].…”

Section: Theorem 2 [Brs]mentioning

confidence: 99%

There Are Not Non-obvious Cyclic Affine-invariant Codes

Bernal

Río

Simón

2009

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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Abstract. We show that an affine-invariant code C of length p m is not permutation equivalent to a cyclic code except in the obvious cases: m = 1 or C is either {0}, the repetition code or its dual.Affine-invariant codes were firstly introduced by Kasami, Lin and Peterson [KLP2] as a generalization of Reed-Muller codes. This class of codes has received the attention of several authors because of its good algebraic and decoding properties [D, BCh, ChL, Ho, Hu]. It is well known that every affine-invariant code can be seen as an ideal of the group algebra of an elementary abelian group in which the group is identified with the standard base of the ambient space. In particular, if C is a code of prime length then C is permutation equivalent to a cyclic code. Other obvious affineinvariant cyclic codes are the trivial code, {0}, the repetition code and the code form by all the even-like words, provided its length is a prime power. In this paper we prove that these are the only affine-invariant codes which are permutation equivalent to a cyclic code.Our main tools are an intrinsical characterization of group codes obtained in [BRS] and a description of the group of permutation automorphisms of non-trivial affine-invariant codes given in [BCh]. These results are reviewed in Section 1, where we also recall the definition and main properties of affine-invariant codes. In Section 2 we prove the main result of the paper.

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Some results on cyclic codes which are invariant under the affine group and their applications

Cited by 124 publications

References 3 publications

The p-adic valuations of Weil sums of binomials

The p-adic valuations of Weil sums of binomials

Limits on the Rate of Locally Testable Affine-Invariant Codes

There Are Not Non-obvious Cyclic Affine-invariant Codes

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