We construct a class of ZprZps-additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic Gilbert-Varshamov bound at 1+ps−r2δ is greater than 12 and the relative distance of the code is convergent to δ, while the rate is convergent to 11+ps−r for 0<δ<11+ps−r and 1≤r

In this article we establish the asymptotic behavior of generating functions related to the exponential sum over finite fields of elementary symmetric functions and their perturbations. This asymptotic behavior allows us to calculate the probability generating function of the probability that the elementary symmetric polynomial of degree k and its perturbations returns β∈Fq where Fq represents the field of q elements. Our study extends many of the results known for perturbations over the binary field to any finite field. In particular, we establish when a particular perturbation is asymptotically balanced over a prime field and provide a construction to find such perturbations over any finite field.

The component-wise or Schur product C⁎C′ of two linear error-correcting codes C and C′ over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C′. When C=C′, we call the product the square of C and denote it C⁎2. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrix-product codes, a general construction that allows to obtain new longer codes from several “constituent” codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u,u+v)-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the squares of other matrix-product codes, for example when the defining matrix is Vandermonde (where the minimum distance is in a certain sense maximal with respect to matrix-product codes).

The k-subset sum problem over finite fields is a classical NP-complete problem. Motivated by coding theory applications, a more complex problem is the higher m-th moment k-subset sum problem over finite fields. We show that there is a deterministic polynomial time algorithm for the m-th moment k-subset sum problem over finite fields for each fixed m when the evaluation set is the image set of a monomial or Dickson polynomial of any degree n. In the classical case m=1, this recovers previous results of Nguyen-Wang (the case m=1,p>2) [22] and the results of Choe-Choe (the case m=1,p=2) [3].

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the works of Heng and Yue (2016) [14] and Tan, Zhou, Tang and Helleseth (2017) [25], we extend Tan, Zhou, Tang and Helleseth's work to obtain a class of optimal 1-weight binary linear codes, new classes of 2-weight and 3-weight p-ary linear codes and a class of 4-weight binary linear codes. The lengths and weight distributions of the t-weight linear codes, where t=1,2,3, are closed-form expressions of Kloosterman sums over finite prime fields, and are completely determined when p=2 and p=3.

Let Fq be the finite field of order q. Let Nq denote the number of solutions to the generalized Markoff-Hurwitz-type equationx1m1+x2m2+⋯+xnmn=bx1t1x2t2⋯xntn with mi,ti∈Z>0 and b∈Fq⁎. Carlitz proposed the problem of finding an explicit formula for Nq for the special form. Let m=m1⋯mn. Cao proved that Nq=qn−1+(−1)n−1 if gcd⁡(∑i=1ntim/mi−m,q−1)=1. In this paper, we obtain an explicit formula for Nq under certain case when gcd⁡(∑i=1ntim/mi−m,q−1)>1. In particular, for the case of mi=ti=2(i=1,…,n), if either gcd⁡(n−1,q−1)=1 or 2n≡4(modq−1), the formula for Nq can be easily deduced. This generalizes Cao's result as well as partially solves Carlitz's problem.

We say that (x,y,z)∈Q3 is an associative triple in a quasigroup Q(⁎) if (x⁎y)⁎z=x⁎(y⁎z). It is easy to show that the number of associative triples in Q is at least |Q|, and it was conjectured that quasigroups with exactly |Q| associative triples do not exist when |Q|>1. We refute this conjecture by proving the existence of quasigroups with exactly |Q| associative triples for a wide range of values |Q|. Our main tools are quadratic Dickson nearfields and the Weil bound on quadratic character sums.

We provide methods and algorithms to construct Hermitian linear complementary dual (LCD) codes over finite fields. We study existence of self-dual basis with respect to Hermitian inner product, and as an application, we construct Euclidean LCD codes by projecting the Hermitian codes over such a basis. Many optimal quaternary Hermitian and ternary Euclidean LCD codes are obtained. Comparisons with classical constructions are made.

Let p>3 be a fixed prime. For a supersingular elliptic curve E over Fp, a result of Ibukiyama tells us that End(E) is a maximal order O(q) (resp. O′(q)) in End(E)⊗Q indexed by a (non-unique) prime q satisfying q≡3mod8 and the quadratic residue (pq)=−1 if 1+π2∉End(E) (resp. 1+π2∈End(E)), where π=((x,y)↦(xp,yp) is the absolute Frobenius. Let qj denote the minimal q for E whose j-invariant j(E)=j and M(p) denote the maximum of qj for all supersingular j∈Fp. Firstly, we determine the neighborhood of the vertex [E] with j∉{0,1728} in the supersingular ℓ-isogeny graph if 1+π2∉End(E) and p>qjℓ2 or 1+π2∈End(E) and p>4qjℓ2: there are either ℓ−1 or ℓ+1 neighbors of [E], each of which connects to [E] by one edge and at most two of which are defined over Fp. We also give examples to illustrate that our bounds are tight. Next, under GRH, we obtain explicit upper and lower bounds for M(p), which were not studied in the literature as far as we know. To make the bounds useful, we estimate the number of supersingular elliptic curves with qjp except p=11 or 23 and M(p)

In this paper we first provide an infinite family of minimal (q−1)-fold blocking sets of size q3 in every affine translation plane of order q2. Next, we focus on the regular Hughes plane H(q2) of order q2. It is well known that π=PG(2,q) is embedded as a Baer subplane in H(q2). Let ℓ be a line of H(q2) having q+1 points in common with π and let us denote by H(q2)ℓ the affine plane obtained from H(q2) by deleting ℓ and all the points incident with ℓ. We exhibit a family of minimal (q−1)-fold blocking sets of size q3 in H(q2)ℓ. Furthermore, we consider the polarity of H(q2) which has q3−q2+q2+1 absolute points and we show that its absolute points form a blocking semioval. Finally, we describe a class of Baer subplanes of H(q2) distinct from π and admitting an automorphism group of order q3(q−1).

We obtain the Singleton bound for poset block codes and define a maximum distance separable poset block code (MDS (P,π)-code) as a code meeting this bound. We extend the concept of I-balls to poset block metric and describe r-perfect and MDS (P,π)-codes in terms of I-perfect codes. As a special case when all the blocks have the same dimension, we establish that MDS (P,π)-codes are same as I-perfect codes for I∈In−km(P). We show that the duality result also holds for this case. Further, we determine the weight enumerator of an MDS (P,π)-code.

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 Fq2. We focus on those with distance d≥q2−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 AFq22 of degree μ+λ defined by polynomials with xμyλ as leading monomial with respect to the DegRevLex term ordering (with y>x). We prove that a zero-dimensional subscheme Z of AFq22 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 Fq2-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.

Linear complementary dual codes were defined by Massey in 1992, and were used to give an optimum linear coding solution for the two user binary adder channel. In this paper, we define the analog of LCD codes over fields in the ambient space with mixed binary and quaternary alphabets. These codes are additive, in the sense that they are additive subgroups, rather than linear as they are not vector spaces over some finite field. We study the structure of these codes and we use the canonical Gray map from this space to the Hamming space to construct binary LCD codes in certain cases. We give examples of such binary LCD codes which are distance-optimal, i.e., they have the largest minimum distance among all binary LCD codes with the same length and dimension.