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 .

Rudolph (1967) introduced one-step majority logic decoding for linear codes derived from combinatorial designs. The decoder is easily realizable in hardware and requires that the dual code has to contain the blocks of so called geometric designs as codewords. Peterson and Weldon (1972) extended Rudolph's algorithm to a two-step majority logic decoder correcting the same number of errors as Reed's celebrated multi-step majority logic decoder. Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved. For a known infinite series of subspace designs the reduction of complexity is exponential.

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.

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.

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.

New linear codes (sometimes optimal) over the finite field with q elements are constructed. In order to do this, an equivalence between the existence of a linear code with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations is used. To reduce the size of the system of equations, the search for solutions is restricted to solutions with special symmetry given by matrix groups. This allows to find more than 400 new codes for the case q=2,3,4,5,7,9.

This correspondence revisits the idea of constructing a binary [mn,mk] code from an [n,k] code over F/sub 2//sup m/ by concatenating the code with a suitable basis representation of F/sub 2//sup m/ over F/sub 2/. We construct two nonequivalent examples of doubly even self-dual binary codes of length 160 which turn out to be of minimum distance 24. This improves the lower bound for this class of codes, whereas the upper bound is given by 28. The construction at hand seems to be of interest beyond this particular example.

In this correspondence, we compute the weight enumerators of various quadratic residue codes over F/sub 2/ and F/sub 3/, together with certain codes of related families like the duadic and the quadratic double circulant codes. We use a parallel algorithm to find the number of codewords of a given (not too high) weight, from which we deduce by usual classical methods for self-dual and formally self-dual codes over F/sub 2/ and F/sub 3/ their associated, previously unknown, weight enumerators. We compute weight enumerators for lengths as high as 152 for binary codes and 96 for ternary codes.

We compute the weight enumerators of various quadratic residue (QR) codes over F/sub 2/ and F/sub 3/, together with certain codes of related families like the duadic codes. We use a parallel algorithm to find the number of codewords of a given (not too high) weight, from which we deduce by usual classical methods for selfdual and isodual codes over F/sub 2/ and F/sub 3/ their associated, previously unknown, weight enumerators. We compute weight enumerators for lengths as high as 152 for binary codes (except for n=138 for which one lacks the number of codewords of weight 34) and 84 for ternary codes.