In this article the wrong figure appeared in Appendix A2; the figure should have appeared as shown below.

We previously proposed the first nontrivial examples of a code having support t-designs for all weights obtained from the Assmus–Mattson theorem and having support $$t'$$ t ′ -designs for some weights with some $$t'>t$$ t ′ > t . This suggests the …

The Korkine–Zolotareff (KZ) reduction and its generalisations, are widely used lattice reduction strategies in communications and cryptography. The KZ constant and Schnorr’s constant were defined by Schnorr in 1987. The KZ constant can be used to …

An $$(r,\ell )$$ ( r , ℓ ) -good polynomial is a polynomial of degree $$r+1$$ r + 1 that is constant on $$\ell $$ ℓ subsets of $$\mathbb F_q$$ F q , each of size $$r+1$$ r + 1 . For any positive integer $$r\le 4$$ r ≤ 4 we provide an $$(r,\ell )$$ …

Motivated by recent works on the butterfly structure, particularly by its generalization introduced by Canteaut et al. (IEEE Trans Inf Theory 63(11):7575–7591, 2017), we first push further the study of permutation polynomials over binary finite …

We consider the class of generalized algebraic geometry codes (GAG codes) formed by two collections of places, with places of the same degree in each collection. We introduce the concept of $$N_1N_2$$ N 1 N 2 -automorphism group of a GAG code in …

Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. A …

We illustrate a general technique to construct towers of fields producing high order elements in $$\mathbb {F}_{q^{2^n}}$$ F q 2 n , for odd q, and in $$\mathbb {F}_{2^{2 \cdot 3^n}}$$ F 2 2 · 3 n , for $$n \ge 1$$ n ≥ 1 . These towers are …

Coding theory and combinatorial t-designs have close connections and interesting interplay. One of the major approaches to the construction of combinatorial t-designs is the employment of error-correcting codes. As we all known, some t-designs …

Delandtsheer and Doyen bounded, in terms of the block size, the number of points of a point-imprimitive, block-transitive 2-design. To do this they introduced two integer parameters m, n, now called Delandtsheer–Doyen parameters, linking the block …

Array codes are the preferred codes for distributed storage, such that different rows in an array are stored at different nodes. Layered codes use a sparse format for stored arrays with a single parity check per column and no other parity checks.

This paper investigates linear-time decoding algorithms for two classes of error-correcting codes. One of the classes is monotone codes which are known as single deletion error-correcting codes, although they are not known to be single …

A cryptographic accumulator is a cryptographic primitive which produces a succinct aggregate of a set of elements. This type of scheme allows to produce a membership proof for each element of the set. In this paper, we propose a code-based …

The original algorithm of D.V. Chudnovsky and G.V. Chudnovsky for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Randriambololona …

Unitary k-designs are probabilistic ensembles of unitary matrices whose first k statistical moments match that of the full unitary group endowed with the Haar measure. In prior work, we showed that the automorphism group of classical $${\mathbb …

We study permutation codes which are groups and all of whose non-identity code elements have the same number of fixed points. It follows that over certain classes of groups such permutation codes exist.

The classification problem for cubic surfaces with 27 lines is concerned with describing a complete set of the projective equivalence classes of such surfaces. Despite a long history of work, the problem is still open. One approach is to use a …

The hull of a linear code C is the intersection of C with its dual $$C^\perp $$ C ⊥ . The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a …

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index bound …

Let $$\mathbb {F}_q$$ F q be the finite field of order $$q=p^m$$ q = p m , where p is a prime, m is a positive integer, and $$\mathcal {R}=\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle $$ R = F q [ u , v ] / ⟨ u 2 - u , v 2 - v , u v - v …