The singleton defect of an [n,k,d] linear code C is defined as s(C)=n−k+1−d. Codes with S(C)=0 are called maximum distance separable (MDS) codes, and codes with S(C)=S(C⊥)=1 are called near maximum distance separable (NMDS) codes. Both MDS codes and NMDS codes have good representations in finite projective geometry. MDS codes over Fq with length n and n-arcs in PG(k−1,q) are equivalent objects. When k=3, NMDS codes of length n are equivalent to (n,3)-arcs in PG(2,q). In this paper, we deal with the NMDS codes with dimension 3. By adding some suitable projective points in maximal arcs of PG(2,q), we can obtain two classes of (q+5,3)-arcs (or equivalently [q+5,3,q+2] NMDS codes) for any prime power q. We also determine the exact weight distribution and the locality of such NMDS codes and their duals. It turns out that the resultant NMDS codes and their duals are both distance-optimal and dimension-optimal locally recoverable codes.

Double Toeplitz（shortly DT） codes are introduced here as a generalization of double circulant codes. The authors show that such a code is isodual, hence formally self-dual（FSD）. FSD codes form a far-reaching generalization of self-dual codes, the most important class of codes of rate one-half. Self-dual DT codes are characterized as double circulant or double negacirculant. Likewise, even binary DT codes are characterized as double circulant. Numerical examples obtained by exhaustive search show that the codes constructed have best-known minimum distance, up to one unit, amongst formally self-dual codes, and sometimes improve on the known values. For q = 2, the authors find four improvements on the best-known values of the minimum distance of FSD codes. Over F 4 an explicit construction of DT codes, based on quadratic residues in a prime field, performs equally well. The authors show that DT codes are asymptotically good over F q . Specifically, the authors construct DT codes arbitrarily close to the asymptotic Varshamov-Gilbert bound for codes of rate one half.

Cyclic codes, as a special type of constacyclic codes, have been extensively studied due to their favorable theoretical and mathematical properties. Very recently, by using the derivative of the Mattson-Solomon polynomials, Huang and Zhang (IEEE Trans Inf Theor 70(4):2395–2410, 2024) studied the cyclic derivative descendants (DDs) and linear DDs of binary extended cyclic codes and proposed the corresponding derivative decoding methods. One objective of this paper is to generalize these conclusions to q-ary extended cyclic codes with group algebra theory. It demonstrates that the cyclic DDs of a q-ary extended cyclic code are the same codes and its linear DDs are equivalent codes. In addition, we show that the relevant results can be generalized to q-ary constacyclic codes and the linear codes generated by Plotkin construction. Our conclusions reveal that the soft-decision decoding method proposed by Huang and Zhang for binary cyclic codes is also applicable to q-ary cyclic codes, q-ary constacyclic codes and the linear codes generated by Plotkin construction.

Subfield codes of linear codes over finite fields have recently received much attention since they can produce optimal codes, which may have applications in secret sharing, authentication codes and association schemes. In this paper, we first present a construction framework of 3-dimensional linear codes C f,g over F qm parameterized by any two functions f , g over F qm , and then study the properties of six types of C f,g , its punctured code C * f,g and their corresponding subfield codes over F q . The classification of C f,g is based on special choices of f , g as trace function, norm function, almost bent function, Boolean bent function or a combination of these functions. For the first two types of C f,g , we explicitly determine the weight distributions and dualities of C f,g , C* f,g and their subfield codes over F q . The remaining four types of C f,g are restricted to q = 2, and the weight distributions and dualities of the subfields code C (q) f,g and C * f,g (q) are completely determined. Most of the resultant linear codes (over F qm or over F q ) have few weights. Some of them are optimal and some have the best-known parameters according to the tables maintained at http://www.codetables.de. In fact, 16 infinite families of optimal linear codes are produced in this paper. As a byproduct, a family of [2 4m-2 , 2 m + 1, 2 4m-3 ] quaternary Hermitian self-orthogonal codes are obtained with m ≥ 2. As an application, we present several infinite families of 2-designs or 3-designs with some of the codes presented in this paper.

The singleton defect of an [ n , k , d ] linear code 𝒞 is defined as s(𝒞)=n-k+1-d . Codes with s(𝒞)=s(𝒞^)=1 are called near maximum distance separable (NMDS) codes. It is known that an [n,3,n-3] NMDS code is equivalent to an ( n , 3)-arc in PG(2, q ). In this paper, by adding some suitable projective points into some known (q+5,3) -arcs in PG(2, q ), we obtain two families of [q+7,3,q+4] NMDS codes for even prime power q and a family of [q+6,3,q+3] NMDS codes for odd prime power q . In addition, when q=2^m and m is odd, by adding m suitable projective points into the maximum arcs in PG(2, q ), we obtain a family of [q+m+2,3,q+m-1] NMDS codes over 𝔽_q , from which we further induce a family of NMDS codes with parameters [q^t+m+2,3,q^t+m-1] over the extension field 𝔽_q^t for any odd integer t . All the resulting NMDS codes in this paper are shown to be linearly inequivalent to the NMDS codes constructed from elliptic curves, and their weight distributions are completely determined.

An [n,k,d] linear code is said to be maximum distance separable (MDS) or almost maximum distance separable (AMDS) if d=n−k+1 or d=n−k, respectively. If a code and its dual code are both AMDS, then the linear code is called a near maximum distance separable (NMDS) code. NMDS codes correspond to some interesting objects in finite geometry and have nice applications in cryptography. The NMDS codes constructed from elliptic curves are referred as elliptic curve NMDS codes. In this paper, by adding two suitable columns to the generator matrices of famous Reed-Solomon codes, we can obtain two families of NMDS codes with parameters [q+3,4,q−1]q (q is a prime power with gcd⁡(q−1,3)=1) and [2m+3,5,2m−2]2m respectively. These NMDS codes are shown to be linearly inequivalent to elliptic curve NMDS codes. Besides, the weight enumerators and locality of these NMDS codes are completely determined. It turns out that the resultant NMDS codes and their dual codes are mostly distance-optimal and dimension-optimal locally recoverable codes.

Locally recoverable codes (LRCs) have been introduced as a family of erasure codes that support the repair of a failed storage node by contacting a small number of other nodes in the cluster. Boosted by their applications in distributed storage, LRCs have attracted a lot of attention in recent literature since the concept of codes with locality r was introduced by Gopalan et al. in 2012. Aiming to recover the data from several concurrent node failures, the concept of r -locality was later generalized as (r, δ ) -locality by Prakash et al. An (r, δ ) -LRCs in which every code symbol has (r, δ ) -locality is said to be optimal if it achieves the Singleton-like bound with equality. In present paper, we are interested in optimal (r, δ ) -LRCs over small fields, more precisely, over quaternary field. We study their parity-check matrices or generator matrices, using the properties of projective space. The classification of optimal quaternary (r,δ ) -LRCs and their explicit code constructions are proposed by examining all possible parameters.

Subfield codes of linear codes over finite fields have recently received a lot of attention, as some of these codes are optimal and have applications in secrete sharing, authentication codes and association schemes. In this paper, the q -ary subfield codes C ( q ) f,g of six different families of linear codes C f,g are presented, respectively. The parameters and weight distribution of the subfield codes and their punctured codes ¯ C ( q ) f,g are explicitly determined. The parameters of the duals of these codes are also studied. Some of the resultant q -ary codes C ( q ) f,g , ¯ C ( q ) f,g and their dual codes are optimal and some have the best known parameters. The parameters and weight enumerators of the first two families of linear codes C f,g are also settled, among which the first family is an optimal two-weight linear code meeting the Griesmer bound, and the dual codes of these two families are almost MDS codes. As a byproduct of this paper, a family of [2 4 m − 2 , 2 m + 1 , 2 4 m − 3 ] quaternary Hermitian self-dual code are obtained with m > 2 . As an application, several infinite families of 2-designs and 3-designs are also constructed with three families of linear codes of this paper.

Double Toeplitz (DT) codes are codes with a generator matrix of the form (I,T) with T a Toeplitz matrix, that is to say constant on the diagonals parallel to the main. When T is tridiagonal and symmetric we determine its spectrum explicitly by using Dickson polynomials, and deduce from there conditions for the code to be LCD. Using a special concatenation process, we construct optimal or quasi-optimal examples of binary and ternary LCD codes from DT codes over extension fields.

Double polycirculant codes are introduced here as a generalization of double circulant codes. When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality, the trivial case of isoduality, can only occur over $$ {\mathbb {F}}_2$$ in the double circulant case. Building on an explicit infinite sequence of irreducible trinomials over $${\mathbb {F}}_2,$$ we show that binary double polycirculant codes are asymptotically good.

The p-rank of a Steiner quadruple system B is the dimension of the linear span of the set of characteristic vectors of blocks of B over GF(p). We derive a formula for the number of different Steiner quadruple systems of order nu and given 2-rank r, r < nu. Our result extends previous work on enumerating Steiner quadruple systems according to the rank of their codes over GF(2), mainly by V.A. Zinoviev and D.V. Zinoviev. (C) 2020 Elsevier B.V. All rights reserved.

In a recent work, Jungnickel, Magliveras, Tonchev, and Wassermann derived an overexponential lower bound on the number of nonisomorphic resolvable Steiner triple systems (STS) of order v, where $$v=3^k$$, and 3-rank $$v-k$$. We develop an approach to generalize this bound and estimate the number of isomorphism classes of resolvable STS (v) of 3-rank $$v-k-1$$ for an arbitrary v of form $$3^kT$$, where T is congruent to 1 or 3 modulo 6.

The p ‐rank of a Steiner triple system (STS) B is the dimension of the linear span of the set of characteristic vectors of blocks of B , over GF ( p ) . We derive a formula for the number of different STSs of order v and given 2 ‐rank r 2 , r 2 < v , and a formula for the number of STSs of order v and given 3 ‐rank r 3 , r 3 < v − 1 . Also, we prove that there are no STSs of 2 ‐rank smaller than v and, at the same time, 3 ‐rank smaller than v − 1 . Our results extend previous study on enumerating STSs according to the rank of their codes, mainly by Tonchev, V.A. Zinoviev, and D.V. Zinoviev for the binary case and by Jungnickel and Tonchev for the ternary case.

Double polycirculant codes are introduced here as a generalization of double circulant codes. They form a special class of quasi-polycyclic codes of index $2.$ When the matrix of the polyshift is a companion matrix of a trinomial, we show that such a code is isodual, hence formally self-dual. Numerical examples show that the codes constructed have optimal or quasi-optimal parameters amongst formally self-dual codes. Self-duality can only occur over $ \F_2$ in the double circulant case. Building on the existence of infinitely many irreducible trinomials over $\F_2$ we show that double polycirculant binary codes satisfy the Varshamov-Gilbert bound for linear codes of rate one half. They are thus asymptotically good.