Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

Let L be an even lattice of odd rank with discriminant group L ′ / L {L^{\prime}/L} , and let α , β ∈ L ′ / L {\alpha,\beta\in L^{\prime}/L} . We prove the Weil bound for the Kloosterman sums S α , β ⁢ ( m , n , c ) {S_{\alpha,\beta}(m,n,c)} of half-integral weight for the Weil Representation attached to L. We obtain this bound by proving an identity that relates a divisor sum of Kloosterman sums to a sparse exponential sum. This identity generalizes Kohnen’s identity for plus space Kloosterman sums with the theta multiplier system.

In this paper, we obtain a new class of $p$-ary binomial bent functions which are determined by Kloosterman sums. The bentness of another three classes of functions is characterized by some exponential sums and some results in \cite{Linian2013} are generalized. Furthermore we obtain, in some special cases, some bent functions are determined by Kloosterman sums.

Here we derive a recursive formula for even-power moments of Kloosterman sums or equivalently for power moments of two-dimensional Kloosterman sums. This is done by using the Pless power-moment identity and an explicit expression of the Gauss sum for Sp(4,q).

Constacyclic codes are well-known generalizations of cyclic and negacyclic codes. Due to their rich algebraic structure, constacyclic codes are used to construct quantum codes and symbol-pair codes. Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> be a finite field with order <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is a positive power of a prime <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula> . Suppose that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is a positive integer and the product of distinct prime factors of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> divides <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> , i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$rad(n)\mid (q-1)$ </tex-math></inline-formula> . In this paper, we explicitly factorize the polynomial <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x^{n}-\lambda $ </tex-math></inline-formula> for each <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda \in {\mathbb {F}}_{q}^{*}$ </tex-math></inline-formula> . As applications, first, we obtain all repeated-root <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> -constacyclic codes and their dual codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$np^{s}$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> ; second, we determine all simple-root LCD cyclic codes and LCD negacyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> ; third, we list all self-dual repeated-root negacyclic codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$np^{s}$ </tex-math></inline-formula> over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathbb {F}}_{q}$ </tex-math></inline-formula> . In contrast to known results, the lengths of constacyclic codes in this paper have more flexible parameters.

On the linear ordering of some classes of negacyclic and cyclic codes and their distance distributions

This article is a survey of several recent applications of methods from analytic number theory to research in coding theory, including results on Kloosterman codes, binary Goppa codes, and prime phase shift sequences. The mathematical methods focus on exponential sums, in particular Kloosterman sums. The interrelationships with the Weil–Carlitz–Uchiyama bound, results on Hecke operators, theorems of Bombieri and Deligne and the Eichler–Selberg trace formula are reviewed.

Cyclic codes are a cornerstone of coding theory. Many studies have extensively explored specific finite prime fields, GF(p). However, a generalized framework for GF(p) remains unexplored. Previous research has enumerated cyclic codes over GF(13), GF(17), GF(19), GF(23), GF(31), and GF(37), deriving significant findings for special cases where n assumes specific forms like $n = p^m$ or $n = a^m \cdot p^m$. Despite these advances, existing studies are often restricted to individual prime fields and lack a unified. framework applicable across all GF(p). This research develops a mathematical framework for counting cyclic codes over any prime field GF(p). The methodology involves the use of rigorous mathematical derivations and proofs to establish the relationship between number of cyclic codes, cyclotomic cosets and the factorization of $x^n - 1$ into irreducible polynomials over GF(p). This framework provides a comprehensive understanding of how the structures of n and p influence the number of cyclic codes. The study concludes that the number of cyclic codes, $\mathcal{N}$, over the prime fields can be expressed as $\mathcal{N} = (p^y + 1)^C$, where C is derived from the number of cyclotomic cosets modulo $k \forall n = k \cdot p^y $. This work contributes to the efficient design of error-correcting codes, strengthens the theoretical foundation for secure communication systems, and bridges theoretical concepts of pure mathematics.

Divisibility properties of classical binary Kloosterman sums

Mean value of some exponential sums and applications to Kloosterman sums

Type 2 constacyclic codes over [formula omitted] of oddly even length

This paper proposes a novel algorithm to find cyclically permutable codes (CPCs) from non-prime-length cyclic codes. In recent years, CPCs have gained increasing attention and have found widespread applications in communication networks and optical communication. A CPC is a block code such that each codeword has full cyclic order and all codewords are cyclically distinct. In this paper, we use the properties of Galois fields to develop an efficient algorithm for finding CPCs from non-prime-length q-ary cyclic codes. More precisely, we propose an effective method to find all codewords with complete cyclic orders from cyclic codes of non-prime code lengths.

Polyadic constacyclic codes over finite fields have been of interest due to their nice algebraic structures, good parameters, and wide applications. Recently, the study of Type-I m-adic constacyclic codes over finite fields has been established. In this paper, a family of Type-II m-adic constacyclic codes is investigated. The existence of such codes is determined using the length of orbits in a suitable group action. A necessary condition and a sufficient condition for a positive integer s to be a multiplier of a Type-II m-adic constacyclic code are determined. Subsequently, for a given positive integer m, a necessary condition and a sufficient condition for the existence of Type-II m-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters.

Let $\mathbb {F}_{p^{m}}$ be a finite field with pm elements for an odd prime p and a positive integer m. In this paper, we aim to construct non-binary quantum codes from repeated root cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ . To achieve this, first we discuss the structure of all maximum distance separable (MDS) cyclic and negacyclic codes of length 4ps over $\mathbb {F}_{p^{m}}$ and their Euclidean and Hermitian duals. Further, we establish a necessary and sufficient condition for these codes to contain their duals. Finally, many new and quantum MDS codes are obtained with the help of CSS and Hermitian constructions from dual containing cyclic and negacyclic codes.

&lt;abstract&gt;&lt;p&gt;In this paper, we propose a way to partition any constacyclic code over a finite field in its equivalence classes according to the algebraic structure of the code. Such a method gives the generalization of cyclically permutable codes (CPCs), which are called constacyclically permutable codes (CCPCs), and it is useful to derive a CCPC from a given constacyclic code. Moreover, we present an enumerative formula for the code size of such a CCPC, with all of the terms being positive integers, and we provide an algebraic method to produce such a CCPC.&lt;/p&gt;&lt;/abstract&gt;

The finite field $\mathbb {F}_{q^\ell }$ of $q^\ell $ elements contains $\mathbb {F}_{q}$ as a subfield. If $\theta \in \mathbb {F}_{q^\ell }$ is of degree $\ell $ over $\mathbb {F}_{q}$ , it can be used to unfold elements of $\mathbb {F}_{q^\ell }$ to vectors in $\mathbb {F}_{q}^\ell $ . We apply the unfolding to the coordinates of all codewords of a cyclic code $\mathcal {C}$ over $\mathbb {F}_{q^\ell }$ of length $n$ . This generates a quasi-cyclic code $\mathcal {Q}$ over $\mathbb {F}_{q}$ of length $n\ell $ and index $\ell $ . We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a generator polynomial $g(x)$ of a cyclic code $\mathcal {C}$ , we present a formula for a generator polynomial matrix for the unfolded code $\mathcal {Q}$ . On the other hand, for any quasi-cyclic code $\mathcal {Q}$ with a reduced generator polynomial matrix $G$ , we provide a necessary and sufficient condition on $G$ that determines whether or not the code $\mathcal {Q}$ can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set $\mathcal {T}$ of the cyclic code $\mathcal {C}$ that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal generator polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.