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Presenting sporadic simple groups as an automorphism groups of designs and graphs is an exciting field in finite group theory. In this paper, with two different methods, we present some new resolvable simple 3-designs with Higman-Sims sporadic simple group HS as the full automorphism group. Also, we classify all block-transitive self-orthogonal designs on 176 points with even block size that admit sporadic simple group HS as an automorphism group. Furthermore, with these methods we construct some new resolvable 3-designs on 36, 40, 120 and 176 points. [ABSTRACT FROM AUTHOR]

The Bose-Chaudhuri-Hocquenghem (BCH) codes are a well-studied subclass of cyclic codes that have found numerous applications in error correction and notably in quantum information processing. They are widely used in data storage and communication systems. A subclass of attractive BCH codes is the narrow-sense BCH codes over the Galois field ${\mathrm {GF}}(q)$ with length $q+1$ , which are closely related to the action of the projective general linear group of degree two on the projective line. Despite its interest, not much is known about this class of BCH codes. This paper aims to study some of the codes within this class and specifically narrow-sense antiprimitive BCH codes (these codes are also linear complementary duals (LCD) codes that have interesting practical recent applications in cryptography, among other benefits). We shall use tools and combine arguments from algebraic coding theory, combinatorial designs, and group theory (group actions, representation theory of finite groups, etc.) to investigate narrow-sense antiprimitive BCH Codes and extend results from the recent literature. Notably, the dimension, the minimum distance of some $q$ -ary BCH codes with length $q+1$ , and their duals are determined in this paper. The dual codes of the narrow-sense antiprimitive BCH codes derived in this paper include almost MDS codes. Furthermore, the classification of ${\mathrm {PGL}}(2, p^{m})$ -invariant codes over ${\mathrm {GF}}(p^{h})$ is completed. As an application of this result, the $p$ -ranks of all incidence structures invariant under the projective general linear group ${\mathrm {PGL}}(2, p^{m})$ are determined. Furthermore, infinite families of narrow-sense BCH codes admitting a 3-transitive automorphism group are obtained. Via these BCH codes, a coding-theory approach to constructing the Witt spherical geometry designs is presented. The BCH codes proposed in this paper are good candidates for permutation decoding, as they have a relatively large group of automorphisms. [ABSTRACT FROM AUTHOR]

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper. [ABSTRACT FROM AUTHOR]

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes $\mathcal {C}_{(q,q+1,4,1)}$ and their dual codes with $q=2^{m}$ and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of $t$ -designs with $t=3,4$. In particular, we prove that codewords of weight 7 in $\mathcal {C}_{(q,q+1,4,1)}$ support 4-designs for odd $m \geqslant 5$ and they support 3-designs for even $m \geqslant 4$ , which provide infinite classes of simple $t$ -designs with new parameters. Another significant class of $t$ -designs we produce in this paper has complementary designs with parameters 4- $(2^{2s+1}+ 1,5,5)$ ; these designs have the smallest index among all the known simple 4- $(q+1,5,\lambda)$ designs derived from codes for prime powers $q$ ; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL $(2,2^{2s+1})$ as the automorphism group constructed by Alltop in 1969. [ABSTRACT FROM AUTHOR]

Linear codes generated by component functions of perfect nonlinear (PN for short) and almost perfect nonlinear (APN for short) functions and the first-order Reed-Muller codes have been an object of intensive study in coding theory. The objective of this paper is to investigate some binary shortened codes of two families of linear codes from APN functions and some $p$ -ary shortened codes associated with PN functions. The weight distributions of these shortened codes and the parameters of their duals are determined. The parameters of these binary codes and $p$ -ary codes are flexible. Many of the codes presented in this paper are optimal or almost optimal. The results of this paper show that the shortening technique is very promising for constructing good codes. [ABSTRACT FROM AUTHOR]

In this paper, a family of $[{q}^{2}-1, 4, {q}^{2}-{q}-2]$ cyclic codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2}-1,{q}^{2}-5,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of cyclic codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Some of the subfield codes are optimal and some have the best known parameters. It is shown that the subfield codes are equivalent to a family of primitive BCH codes and thus the parameters of the BCH codes are solved. The duals of the subfield codes are also optimal with respect to the sphere-packing bound. A family of $[{q}^{2}, 4, {q}^{2}-{q}-1]$ linear codes over ${\mathbb F}_{{q}}$ meeting the Griesmer bound is presented. Their duals are $[{q}^{2},{q}^{2}-4,4]$ almost MDS codes and are optimal with respect to the sphere-packing bound. The ${q}_{0}$ -ary subfield codes of this family of linear codes are also investigated, where ${q}_{0}$ is any prime power such that q is power of ${q}_{0}$. Five infinite families of 2-designs are also constructed with three families of linear codes of this paper. [ABSTRACT FROM AUTHOR]

Combinatorial t-designs have nice applications in coding theory, finite geometries and engineering areas. t-designs can be constructed from image sets of a fixed size of some special polynomials. This paper constructs t-designs from the quadratic polynomial x 4 + x 3 over GF (2 n) and determine their parameters. We yield 2- 2 n , 3 · 2 n - 2 , 3 · 2 n - 2 3 · 2 n - 2 - 1 designs for n even and 3- 2 n , 2 n - 1 , 2 n - 1 2 n - 2 - 1 designs for n odd. [ABSTRACT FROM AUTHOR]

There are two standard approaches to the construction of t-designs. The first one is based on permutation group actions on certain base blocks. The second one is based on coding theory. The objective of this paper is to give a spectral characterisation of all t-designs by introducing a characteristic Boolean function of a t-design. The spectra of the characteristic functions of (n−2)/2-(n,n/2,1) Steiner systems are determined and properties of such designs are proved. Delsarte's characterisations of orthogonal arrays and t-designs, which are two special cases of Delsarte's characterisation of T-designs in association schemes, are slightly extended into two spectral characterisations. Another characterisation of t-designs by Delsarte and Seidel is also extended into a spectral one. These spectral characterisations are then compared with the new spectral characterisation of this paper. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are very important in both theory and practice. There is a classical construction of a family of $[{2^{m}+1, 2u-1, 2^{m}-2u+3}]$ MDS codes for $1 \leq u \leq 2^{m-1}$ , which are cyclic, reversible and BCH codes over ${\mathrm {GF}}(2^{m})$. The objective of this paper is to study the quaternary subfield subcodes and quaternary subfield codes of a subfamily of the MDS codes for even $m$. A family of quaternary cyclic codes is obtained. These quaternary codes are distance-optimal in some cases and very good in general. Furthermore, two infinite families of 3-designs from these quaternary codes and their duals are presented. [ABSTRACT FROM AUTHOR]

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 ′ -designs for some weights with some t ′ > t . This suggests the possibility of generalizing the Assmus–Mattson theorem, which is very important in design and coding theory. In the present paper, we generalize this example as a strengthening of the Assmus–Mattson theorem along this direction. As a corollary, we provide a new characterization of the extended Golay code G 24 . [ABSTRACT FROM AUTHOR]

In this paper, we determine all simple block-transitive 3 - and 4 -designs on 2 8 points admitting the group S 6 (2) as the automorphism group. Thereby, we found 15 new 3 -designs and two new 4 -designs with the automorphism group S 6 (2) up to isomorphism. Some of these designs are self-complementary and resolvable. [ABSTRACT FROM AUTHOR]

Boolean functions, coding theory and $t$ -designs have close connections and interesting interplay. A standard approach to constructing $t$ -designs is the use of linear codes with certain regularity. The Assmus-Mattson Theorem and the automorphism groups are two ways for proving that a code has sufficient regularity for supporting $t$ -designs. However, some linear codes hold $t$ -designs, although they do not satisfy the conditions in the Assmus-Mattson Theorem and do not admit a $t$ -transitive or $t$ -homogeneous group as a subgroup of their automorphisms. The major objective of this paper is to develop a theory for explaining such codes and obtaining such new codes and hence new $t$ -designs. To this end, a general theory for punctured and shortened codes of linear codes supporting $t$ -designs is established, a generalized Assmus-Mattson theorem is developed, and a link between 2-designs and differentially $\delta $ -uniform functions and 2-designs is built. With these general results, binary codes with new parameters and explicit weight distributions are obtained, new 2-designs and Steiner system $S(2, 4, 2^{n})$ are produced in this paper. [ABSTRACT FROM AUTHOR]

Combinatorial t‐designs have nice applications in coding theory, finite geometries, and several engineering areas. A classical method for constructing t‐designs is by the action of a permutation group that is t‐transitive or t‐homogeneous on a point set. This approach produces t‐designs, but may not yield (t+1)‐designs. The objective of this paper is to study how to obtain 3‐designs with 2‐transitive permutation groups. The incidence structure formed by the orbits of a base block under the action of the general affine groups, which are 2‐transitive, is considered. A characterization of such incidence structure to be a 3‐design is presented, and a sufficient condition for the stabilizer of a base block to be trivial is given. With these general results, infinite families of 3‐designs are constructed by employing almost perfect nonlinear functions. Some 3‐designs presented in this paper give rise to self‐dual binary codes or linear codes with optimal or best parameters known. Several conjectures on 3‐designs and binary codes are also presented. [ABSTRACT FROM AUTHOR]

Coding theory and t-designs have close connections and interesting interplay. In this paper, we first introduce a class of ternary linear codes and study their parameters. We then focus on their three-weight subcodes with a special weight distribution. We determine the weight distributions of some shortened codes and punctured codes of these three-weight subcodes. These shortened and punctured codes contain some codes that have the same parameters as the best ternary linear codes known in the database maintained by Markus Grassl at http://www.codetables.de/. These three-weight subcodes with a special weight distribution do not satisfy the conditions of the Assmus–Mattson theorem and do not admit 2-transitive or 2-homogeneous automorphism groups in general. By employing the theory of projective geometries and projective generalized Reed–Muller codes, we prove that they still hold 2-designs. We also determine the parameters of these 2-designs. This paper mainly confirms some recent conjectures of Ding and Li regarding Steiner systems and 2-designs from a special type of ternary projective codes. [ABSTRACT FROM AUTHOR]

There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. Recently, several infinite families of 4-designs from linear codes were discovered. But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. In this paper, we construct eight 5-designs from some lifted codes of some known linear codes. • There are many infinite families of 2-designs and 3-designs supported by linear codes in the literature. • Recently, several infinite families of 4-designs from linear codes were discovered. • But no infinite family of linear codes supporting an infinite family of nontrivial simple 5-designs is reported in the literature. • It is not easy to construct 5-designs from sporadic codes. Only a small number of sporadic 5-designs supported by linear codes are available in the literature. • In this paper, we construct eight new 5-designs from some lifted codes of some known linear codes. [ABSTRACT FROM AUTHOR]

Let q = 2 m . The projective general linear group PGL (2 , q) acts as a 3-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over GF (2 h) that are invariant under PGL (2 , q) are trivial codes: the repetition code, the whole space GF (2 h) 2 m + 1 , and their dual codes. As an application of this result, the 2-ranks of the (0,1)-incidence matrices of all 3 - (q + 1 , k , λ) designs that are invariant under PGL (2 , q) are determined. The second objective is to present two infinite families of cyclic codes over GF (2 m) such that the set of the supports of all codewords of any fixed nonzero weight is invariant under PGL (2 , q) , therefore, the codewords of any nonzero weight support a 3-design. A code from the first family has parameters [ q + 1 , q - 3 , 4 ] q , where q = 2 m , and m ≥ 4 is even. The exact number of the codewords of minimum weight is determined, and the codewords of minimum weight support a 3- (q + 1 , 4 , 2) design. A code from the second family has parameters [ q + 1 , 4 , q - 4 ] q , q = 2 m , m ≥ 4 even, and the minimum weight codewords support a 3- (q + 1 , q - 4 , (q - 4) (q - 5) (q - 6) / 60) design, whose complementary 3- (q + 1 , 5 , 1) design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over GF (q) that can support a 3- (q + 1 , q - 4 , (q - 4) (q - 5) (q - 6) / 60) design is proved, and it is shown that the designs supported by the codewords of minimum weight in the codes from the second family of codes meet this bound. [ABSTRACT FROM AUTHOR]

An $[{n}, {k}, {n}-{k}+1]$ linear code is called an MDS code. An $[{n}, {k}, {n}-{k}]$ linear code is said to be almost maximum distance separable (almost MDS or AMDS for short). A code is said to be near maximum distance separable (near MDS or NMDS for short) if the code and its dual code both are almost maximum distance separable. The first near MDS code was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. This ternary code holds 4-designs, and its extended code holds a Steiner system ${S}(5, 6, 12)$ with the largest strength known. In the past 70 years, sporadic near MDS codes holding t-designs were discovered and a lot of infinite families of near MDS codes over finite fields were constructed. However, the question as to whether there is an infinite family of near MDS codes holding an infinite family of t-designs for $ {t}\geq 2$ remains open for 70 years. This paper settles this long-standing problem by presenting an infinite family of near MDS codes over $ {GF}(3^{ {s}})$ holding an infinite family of 3-designs and an infinite family of near MDS codes over $ {GF}(2^{2 {s}})$ holding an infinite family of 2-designs. The subfield subcodes of these two families of codes are also studied, and are shown to be dimension-optimal or distance-optimal. [ABSTRACT FROM AUTHOR]

In this paper we construct transitive t-designs from the linear groups L(2, q), q ≤ 23. Thereby we classify t-designs, t ≥ 2, admitting a transitive action of the linear groups L(2, q), q ≤ 23, up to 35 points and obtained numerous transitive designs, for 36 ≤ v ≤ 55. In many cases we proved the existence of t-designs with certain parameter sets. Among others we constructed t-designs with parameters 2-(55, 10, 4), 3-(24, 11, 495), 3-(24, 12, 5m), m 2 f11, 12, 22, 33, 44, 66, 132g. Furthermore, we constructed strongly regular graphs admitting a transitive action of the linear groups L(2, q), q ≤ 23. [ABSTRACT FROM AUTHOR]

Quantum jump codes are quantum codes that correct errors caused by quantum jumps. A spontaneous emission error design (SEED) was introduced by Beth et al. in 2003 to construct quantum jump codes. In this paper, we study the existence of 3-SEEDs from PSL(2, q) or PGL(2, q). By doing this, a large number of 3- [ABSTRACT FROM AUTHOR]

The projective special linear group ${\mathrm {PSL}}_{2}(n)$ is 2-transitive for all primes $n$ and 3-homogeneous for $n \equiv 3 \pmod {4}$ on the set $\{0,1, \ldots, n-1, \infty \}$. It is known that the extended odd-like quadratic residue codes are invariant under ${\mathrm {PSL}}_{2}(n)$. Hence, the extended quadratic residue codes hold an infinite family of 2-designs for primes $n \equiv 1 \pmod {4}$ , an infinite family of 3-designs for primes $n \equiv 3 \pmod {4}$. To construct more $t$ -designs with $t \in \{2, 3\}$ , one would search for other extended cyclic codes over finite fields that are invariant under the action of ${\mathrm {PSL}}_{2}(n)$. The objective of this paper is to prove that the extended quadratic residue binary codes are the only nontrivial extended binary cyclic codes that are invariant under ${\mathrm {PSL}}_{2}(n)$. [ABSTRACT FROM AUTHOR]

The combinatorial aspects of quantum codes were demonstrated in the study of decay processes of certain quantum systems used in the newly emerging field of quantum computing. Among them, the configuration of t-spontaneous emission error design ( t-SEED) was proposed to correct errors caused by quantum jumps. The number of designs (dimension) in a t-SEED corresponds to the number of orthogonal basis states in a quantum jump code. In this paper the upper and lower bounds on the dimensions of 3- $$(v,\,4;\,m)$$ SEEDs are studied and the necessary and sufficient conditions for 3-SEEDs attaining the upper bounds are described. Several new combinatorial constructions are presented for general t-SEEDs and lots of t-SEEDs of new parameters with $$t\ge 3$$ are shown to exist. [ABSTRACT FROM AUTHOR]

In this paper we propose unreduced balanced incomplete block designs which are multiple balanced incomplete block designs of resolvable type designs with no repeated blocks and also have quasi - symmetric structure with k intersection numbers between the blocks with illustrations. [ABSTRACT FROM AUTHOR]

In "Infinite families of near MDS codes holding t -designs, IEEE Trans. Inform. Theory, 2020, 66(9), pp. 5419-5428", Ding and Tang made a breakthrough in constructing the first two infinite families of NMDS codes holding 2-designs or 3-designs. Up to now, there are only a few known infinite families of NMDS codes holding t -designs in the literature. The objective of this paper is to construct new infinite families of NMDS codes holding t -designs. We determine the weight enumerators of the NMDS codes and prove that the NMDS codes hold 2-designs or 3-designs. Compared with known t -designs from NMDS codes, ours have different parameters. Besides, several infinite families of optimal locally recoverable codes are also derived via the NMDS codes. [ABSTRACT FROM AUTHOR]

In this paper, we study the automorphism groups of 4-design. Let 4 - (q2 - 1, k, 4) be a non trivial design D = (P, B), n ⩾ 1, q = 32n+1 and G contained in Aut(D) admitting a block-transitively on D. If Soc(G) ≃ GUn(q), then G is not flag-transitive. [ABSTRACT FROM AUTHOR]

We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by 12 and quaternary near-extremal Hermitian self-dual codes of length divisible by 6. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length 12m for m = 3 , 4 , 5 , 6 . Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length 6m for m = 4 , 5 , 6 . [ABSTRACT FROM AUTHOR]

Hyperovals in projective planes turn out to have a link with t‐designs. Motivated by an unpublished work of Lonz and Vanstone, we present a construction for t‐designs and s‐resolvable t‐designs from hyperovals in projective planes of order 2n. We prove that the construction works for t≤5. In particular, for t=5 the construction yields a family of 5‐(2n+2,8,70(2n−2−1)) designs. For t=4 numerous infinite families of 4‐designs on 2n+2 points with block size 2k can be constructed for any k≥4. The construction assumes the existence of a 4‐(2n−1+1,k,λ) design, called the indexing design, including the complete 4‐(2n−1+1,k,(2n−1−3k−4)) design. Moreover, we prove that if the indexing design is s‐resolvable, then so is the constructed design. As a result, many of the constructed designs are s‐resolvable for s=2,3. We include a short discussion on the simplicity or non‐simplicity of the designs from hyperovals. [ABSTRACT FROM AUTHOR]

Abstract: A pair { T 0 , T 1 } of disjoint collections of k‐subsets (blocks) of a set V of cardinality v is called a t‐ ( v , k ) trade or simply a t‐trade if every t‐subset of V is included in the same number of blocks of T0 and T1. The cardinality of T0 is called the volume of the trade. Using the weight distribution of the Reed–Muller code, we prove the conjecture that for every i from 2 to t, there are no t‐trades of volume greater than 2 t + 1 − 2 i and less than 2 t + 1 − 2 i − 1 and derive restrictions on the t‐trade volumes that are less than 2 t + 1 + 2 t − 1. [ABSTRACT FROM AUTHOR]

Looking at incidence matrices of t- $$(v,\,k,\,\lambda )$$ designs as $$v \times b$$ matrices with two possible entries, each of which indicates incidences of a t-design, we introduce the notion of a c-mosaic of designs, having the same number of points and blocks, as a matrix with c different entries, such that each entry defines incidences of a design. In fact, a $$v \times b$$ matrix is decomposed in c incidence matrices of designs, each denoted by a different colour, hence this decomposition might be seen as a tiling of a matrix with incidence matrices of designs as well. These mosaics have applications in experiment design when considering a simultaneous run of several different experiments. We have constructed infinite series of examples of mosaics and state some probably non-trivial open problems. Furthermore we extend our definition to the case of q-analogues of designs in a meaningful way. [ABSTRACT FROM AUTHOR]

We give a survey on various design theories from the viewpoint of algebraic combinatorics. We will start with the following themes. Our emphasis will be focused on the proposal to study relative t-designs, mostly tight relative t-designs, in known classical examples of P- and Q-polynomial association schemes. We relate our study with the representation theoretical aspect of the relevant association schemes and permutation groups, due to Charles Dunkl and Dennis Stanton and others. We propose several open problems, which seem to play a key role in this research direction. We also put emphasis on the future research directions in this research area and not on presenting the details of the established results on the study of design theory. In particular, we propose to study t-designs in each shell of these classical P- and Q-polynomial association schemes. In general, each shell is not Q-polynomial. It may be even non-commutative in some cases. The importance of the use of Terwilliger algebras in the study of relative t-designs in such association schemes will also be highlighted. [ABSTRACT FROM AUTHOR]

We provide general criteria for orthogonal arrays and t-designs on equitable partitions of a hypercube [ABSTRACT FROM AUTHOR]