Huffman (2013) [12] studied Fq-linear codes over Fqm and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative Fq-algebra. An Fq-linear code over S of length n is an Fq-submodule of Sn. In this paper, we study Fq-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters [3m−1,3m−1−2m,4] based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over F3m.

The second generalized GK function fields Kn are a recently found family of maximal function fields over the finite field with q2n elements, where q is a prime power and n≥1 an odd integer. In this paper we construct many new maximal function fields by determining various Galois subfields of Kn. In case gcd⁡(q+1,n)=1 and either q is even or q≡1(mod4), we find a complete list of Galois subfields of

In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that(6,1)n=[6,1]n=(3,2)n=[3,2]n=0 and(4,2)n=(8,8)n=(3,3)n=(21,112)n=0 as conjectured by Sun, where(c,d)n=|(i2+cij+dj2n)|1≤i,j≤n−1 and[c,d]n=|(i2+cij+dj2n)|0≤i,j≤n−1 with (⋅n) the Jacobi symbol. We also prove that (10,9)p=0 for any prime p≡5(mod12), and [5

It is known that the dual of a weakly regular bent function is again weakly regular. On the other hand, the dual of a non-weakly regular bent function may not even be a bent function. In 2013, Çesmelioğlu, Meidl and Pott pointed out that the existence of a non-weakly regular bent function having weakly regular bent dual is an open problem. In this paper, we prove that for an odd prime p and n∈Z+, if

A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular, we show that primitive sets in the function field have lower density zero by showing that the sum ∑a∈A1qdeg⁡adeg⁡a, an analogue of a sum considered by Erdős, is uniformly

A locally recoverable (LRC) code is a code over a finite field Fq such that any erased coordinate of a codeword can be recovered from a small number of other coordinates in that codeword. We construct LRC codes correcting more than one erasure, which are subfield-subcodes of some J-affine variety codes. For these LRC codes, we compute localities (r,δ) that determine the minimum size of a set R‾ of

In this paper, a necessary and sufficient condition for the homogeneous distance on an arbitrary finite commutative principal ideal ring to be a metric is obtained. We completely characterize the lower bound of homogeneous distances of matrix product codes over any finite principal ideal ring where the homogeneous distance is a metric. Furthermore, the minimum homogeneous distances of the duals of

Up to linear transformations, we obtain a classification of permutation polynomials (PPs) of degree 8 over F2r with r>3. By Bartoli et al. (2017) [1], a polynomial f of degree 8 over F2r is exceptional if and only if f−f(0) is a linearized PP, which has already been classified. So it suffices to search for non-exceptional PPs of degree 8 over F2r, which exist only when r⩽9 by a previous result. This

In this paper, we generalize the notion of functional graph. Specifically, given an equation E(X,Y)=0 with variables X and Y over a finite field Fq of odd characteristic, we define a digraph by choosing the elements in Fq as vertices and drawing an edge from x to y if and only if E(x,y)=0. We call this graph as equational graph. In this paper, we study the equational graph when choosing E(X,Y)=(Y2−f(X))(λY2−f(X))

We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree k over Fq is induced by an action from a permutation polynomial of Fqk with coefficients in Fq. The dynamics of these permutations of irreducible polynomials of degree k over Fq, such as fixed points and cycle lengths

A congruence on a conjecture of van Hamme is established. This result confirms a particular case of a congruence conjecture of Swisher.

A function f(x) from the finite field GF(pn) to itself is said to be differentially δ-uniform when the maximum number of solutions x∈GF(pn) of f(x+a)−f(x)=b for any a∈GF(pn)⁎ and b∈GF(pn) is equal to δ. Let p=3 and d=3n−3. When n>1 is odd, the power mapping f(x)=xd over GF(3n) was proved to be differentially 2-uniform by Helleseth, Rong and Sandberg in 1999. For even n, they showed that the differential

We compute the second moment in the family of quadratic Dirichlet L–functions with prime conductors over Fq[x] when the degree of the discriminant goes to infinity, obtaining one of the lower order terms. We also obtain an asymptotic formula with the leading order term for the mean value of the derivatives of L–functions associated to quadratic twists of a fixed elliptic curve over Fq(t) by monic irreducible

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

Let Na be the number of solutions to the equation x2k+1+x+a=0 in F2n where gcd⁡(k,n)=1. In 2004, by Bluher [2] it was known that possible values of Na are only 0, 1 and 3. In 2008, Helleseth and Kholosha [13] found criteria for Na=1 and an explicit expression of the unique solution when gcd⁡(k,n)=1. In 2010 [14], the extended version of [13], they also got criteria for Na=0,3. In 2014, Bracken, Tan

The study of permutation polynomials over finite fields has attracted many scholars' attentions due to their wide applications and there are several different forms of permutations over finite fields. However, there is little literature on the relation between different forms of permutations. In this paper, we find an equivalent relation between permutation polynomials of the form f(x)=xh(x2n−1) over

Planar polynomials of type fa,b(x)=ax22m+1+bx2m+1, a,b∈F23m⁎ are investigated. In particular, all the possible pairs (a,b)∈(F23m⁎)2 for which fa,b(x) is planar are determined.

We prove that for an arbitrary ε>0 and any multiplicative subgroup Γ⊆Fp, 1≪|Γ|⩽p2/3−ε there are no sets B, C⊆Fp with |B|,|C|>1 such that Γ=B+C. Also, we obtain that for 1≪|Γ|⩽p6/7−ε and any ξ≠0 there is no a set B such that ξΓ+1=B/B.

In this paper we study a new construction of differentially 4-uniform permutations from known ones and the inverse function. We focus on constructing methods of [20]. We split a finite field into its subfield and remainder, and, we choose known differentially 4-uniform permutations over the subfield and the inverse function over the entire field. As a result, we obtain two families of differentially

We study restriction problem in vector spaces over finite fields. We obtain finite field analogue of Mockenhaupt-Mitsis-Bak-Seenger restriction theorem, and we show that the range of the exponentials is sharp.

We consider a generalisation of a conjecture by Patterson and Wiedemann from 1983 on the Hamming distance of a function from Fqn to Fq to the set of affine functions from Fqn to Fq. We prove the conjecture for each q such that the characteristic of Fq lies in a subset of the primes with density 1 and we prove the conjecture for all q by assuming the generalised Riemann hypothesis. Roughly speaking

Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then (r,δ)-locally repairable codes ((r,δ)-LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r-LRC can be viewed as an (r,2)-LRC. An (r,δ)-LRC is called optimal if it achieves the Singleton-type

Factorization of various types of polynomials over a finite field Fq is a classical problem. However factorization of permutation polynomials of Fq was not studied previously. Here we present results on the degrees of the irreducible factors of a large class of permutation polynomials.

It is known from the CSS code construction that an [[m,2k−m,≥d]]q stabilizer code can be obtained from a (Euclidean) dual-containing [m,k,d]q code. In [5], Blackmore and Norton introduced an interesting code called matrix-product code, which is very useful in constructing new quantum codes of large lengths. Recently, Galindo et al. [16] constructed several classes of stabilizer codes from the dual-containing

In this paper, we construct an infinite family of q−12-ovoids of the generalized quadrangle Q(4,q), for q≡1(mod4) and q>5. Together with [3] and [11], this establishes the existence of q−12-ovoids in Q(4,q) for each odd prime power q.

A polynomial f(x) over a field K is called stable if all of its iterates are irreducible over K. In this paper, we study the stability of trinomials over finite fields. We show that if f(x) is a trinomial of even degree over the binary field F2, then f(x) is not stable. We prove similar results for some families of monic trinomials over finite fields of odd characteristic. We also study the stability

In this paper, we present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). By these characterizations, we obtain some enumeration formulas of SR and SCR irreducible factors of xn−λ, λ∈Fq⁎, over Fq, which are just open questions posed by Boripan et al. (2019). We also count the numbers of Euclidean and Hermitian

For prime powers q and q+ε where ε∈{1,2}, an affine resolvable design from Fq and Latin squares from Fq+ε yield a set of symmetric designs if ε=2 and a set of symmetric group divisible designs if ε=1. We show that these designs derive commutative association schemes, and determine their eigenmatrices.

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

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

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

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

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

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

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

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

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)

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

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

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