Abstract In this article, we construct a Steiner system with the parameters S (3, 6, 42), settling one of the smallest open parameter sets of Steiner 3-designs. Furthermore, we establish the existence of rotational Steiner quadruple systems on 46 and 92 points. Our construction method is based on extending Steiner 2-designs using prescribed extension groups. We also consider extensions to designs of higher strength. The article includes a table and a discussion of the status of all admissible parameters for Steiner 3-designs on at most 50 points.

Abstract Combinatorial designs have been studied for nearly 200 years. 50 years ago, Cameron, Delsarte, and Ray-Chaudhury started investigating their q-analogs, also known as subspace designs or designs over finite fields. Designs can be defined analogously in finite classical polar spaces, too. The definition includes the m-regular systems from projective geometry as the special case where the blocks are generators of the polar space. The first nontrivial such designs for $$t > 1$$ t > 1 were found by De Bruyn and Vanhove in 2012, and some more designs appeared recently in the PhD thesis of Lansdown. In this article, we investigate the theory of classical and subspace designs for applicability to designs in polar spaces, explicitly allowing arbitrary block dimensions. In this way, we obtain divisibility conditions on the parameters, derived and residual designs, intersection numbers and an analog of Fisher’s inequality. We classify the parameters of symmetric designs. Furthermore, we conduct a computer search to construct designs of strength $$t=2$$ t = 2 , resulting in designs for more than 140 previously unknown parameter sets in various classical polar spaces over $$\mathbb {F}_2$$ F 2 and $$\mathbb {F}_3$$ F 3 .

Abstract Strongly walk regular graphs (SWRGs or s-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length 2 are replaced by paths of length s. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an s-SWRG has at most 4 distinct eigenvalues $$k> \theta _1> \theta _2 > \theta _3$$ k > θ 1 > θ 2 > θ 3 , and that the triple $$(\theta _1, \theta _2, \theta _3)$$ ( θ 1 , θ 2 , θ 3 ) satisfies a certain homogeneous polynomial equation of degree $$s - 2$$ s - 2 (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $$s = 5$$ s = 5 and $$s = 7$$ s = 7 , there are only the obvious solutions, and we conjecture this to remain true for all (odd) $$s \ge 9$$ s ≥ 9 .

We classify all $q$-ary $\Delta$-divisible linear codes which are spanned by codewords of weight $\Delta$. The basic building blocks are the simplex codes, and for $q=2$ additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight $4$ have been classified, which is the case $q=2$ and $\Delta=4$ of our classification. As an application, we give an alternative proof of a theorem of Liu on binary $\Delta$-divisible codes of length $4\Delta$ in the projective case.

Abstract 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 point P is called an $$\alpha $$ α -point of D if the derived design of D in P is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non- $$\alpha $$ α -point. For the binary case $$q = 2$$ q = 2 , Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non- $$\alpha $$ α -points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of $$\alpha $$ α -points implies the existence of a partition of the symplectic generalized quadrangle W(q) into spreads. As a consequence, the statement of Heden and Sissokho is generalized to all primes q and all even values of q.

It is shown that there does not exist a projective triply-even binary code of length 59. This settles the last open length for projective triply-even binary codes, which therefore exist precisely for the lengths 15, 16, 30, 31, 32, 45-51, and ≥ 60.

In this article, the effective lengths of all qr-divisible linear codes over Fqwith a non-negative integer r are determined. For that purpose, the Sq(r)-adic expansion of an integer n is introduced. It is shown that there exists a qr-divisible Fq-linear code of effective length n if and only if the leading coefficient of the Sq(r)-adic expansion of n is non-negative. Furthermore, the maximum weight of a qr-divisible code of effective length n is at most σqr, where σ denotes the cross-sum of the Sq(r)-adic expansion of n. This result has applications in Galois geometries. A recent theorem of Nästase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.

Using integer linear programming and table-lookups, we prove that there is no binary linear [1988, 12, 992] code. As a by-product, the non-existence of binary linear codes with the parameters [324, 10, 160], [356, 10, 176], [772, 11, 384], and [836, 11, 416] is shown. Our work is motivated by the recent construction of the extended dualized Kerdock code K6*, which is a Z4-linear code having a non-linear binary Gray image with the parameters 1988, 212,992. By our result, the code K6* can be added to the small list of Z4-codes for which it is known that the Gray image is better than any binary linear code.

We introduce complementary information set codes of higher order. A binary linear code of length tk and dimension k is called a complementary information set code of order t (t-CIS code for short) if it has t pairwise disjoint information sets. The duals of such codes permit to reduce the cost of masking cryptographic algorithms against side-channel attacks. As in the case of codes for error correction, given the length and the dimension of a t-CIS code, we look for the highest possible minimum distance. In this paper, this new class of codes is investigated. The existence of good long CIS codes of order 3 is derived by a counting argument. General constructions based on cyclic and quasi-cyclic codes and on the building up construction are given. A formula similar to a mass formula is given. A classification of 3-CIS codes of length ≤ 12 is given. Nonlinear codes better than linear codes are derived by taking binary images of Z4-codes. A general algorithm based on Edmonds' basis packing algorithm from matroid theory is developed with the following property: given a binary linear code of rate 1/t, it either provides t disjoint information sets or proves that the code is not t-CIS. Using this algorithm, all optimal or best known [tk, k] codes, where t = 3, 4, . . . , 256 and 1≤ k ≤⌊256/t⌋ are shown to be t-CIS for all such k and t, except for t = 3 with k = 44 and t = 4 with k = 37.

It is shown that there is no self-dual \BBZ4-linear code whose Gray image has the parameters (72,236,16).

A fast method to compute the minimum Lee weight and the symmetrized weight enumerator of extended quadratic residue codes (XQR-codes) over the ring Z4is developed. Our approach is based on the classical Brouwer-Zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self-duality of the Z4-linear XQR-codes as well as the projection to the binary XQR-codes. As a result, the hitherto unknown minimum Lee distances of all Z4-linear XQR-codes of lengths between 72 and 104 and the minimum Euclidean distances for the lengths 72, 80, and 104 are computed. It turns out that the binary Gray image of the Z4-linear XQR-codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality. Furthermore, the Z4-linear XQR-code of length 80 is a new example of an extremal Z4-linear typeII code. Additionally, we give the symmetrized weight enumerator of the Z4-linear XQR-codes of lengths 72 and 80, and we correct the weight enumerators of the Z4-linear XQR-code of length 48 given by Pless and Qian and Bonnecaze et al.

In this paper, we construct new binary singly even self-dual codes with larger minimum weights than the previously known singly even self-dual codes for several lengths. Several known construction methods are used to construct the new self-dual codes.

The class of the quadratic residue codes (QR-codes) over the ring Zopf4contains very good Zopf4-linear codes. It is well known that the Gray images of the QR-codes over Zopf4of length 8, 32 and 48 are non-linear binary codes of higher minimum Hamming distance than comparable known linear codes. The QR-Code of length 48 is also the largest one whose exact minimum Lee distance was known. We developed a fast algorithm to compute the minimum Lee distance of QR-codes over Zopf4, and applied it to all Zopf4-linear QR-codes up to length 98. The QR-code of length 80 has minimum Lee distance 26. Thus it is a new example of a Zopf4-linear code which is better than any known comparable linear code.