Solomon-Stiffler codes and Belov codes are two wellknown families of Griesmer codes that have recently attracted significant attention in recent coding-theoretic literature, due to their optimality and special algebraic structures. In this paper, we investigate their subfield codes and hull dimensions. Firstly, we establish the parameters of Solomon-Stiffler and Belov codes by means of exponential sums. Following the approach of Hyun et al. (IEEE Trans. Inf. Theory, 71(6): 4267-4283, 2025), we determine the exact parameters of the subfield codes of Solomon-Stiffler codes and Belov codes; for the non-projective case, the parameters are fixed, while for the projective case, they depend on the number of distinct types of generator matrices of the mutually disjoint subspaces. We also derive an explicit formula for the weight enumerators of Solomon-Stiffler codes. Secondly, we characterize the hull dimensions of Solomon-Stiffler codes and Belov codes, thereby extending the results of Shi et al. (J. Combin. Theory, Ser. A, 214: 106027, 2025) on the self-orthogonality of binary Solomon-Stiffler codes and Belov codes. As a consequence, we also obtain several families of self-orthogonal codes.

In this paper, we mainly investigate profound interconnections between combinatorial designs, linear codes, and Boolean functions. Firstly, we present a generic construction method for designs derived from Boolean functions and give a new concept of non-symmetric designs with the triple symmetric difference property (TSDP). Secondly, we provides an alternative proof for addition designs derived from plateaued functions, which need not be simple or symmetric. We characterize simple 2-designs on 2m−rpoints arising fromr-plateaued functions inmvariables, and show that addition designs from such functions with no nonzero linear structure satisfy the TSDP but not the double one, yielding non-symmetric simple 2-designs. Thirdly, we primarily explore the equivalence relationships between designs, linear codes, and plateaued functions. These investigations help resolve two open problems posed by Ding and Tang (Designs from Linear codes, Singapore: World Scientific, 2022: Problems 14.20, 14.23).We also compute the automorphism groups of addition designs ofr-plateaued functions and of the linear codes of addition designs. This work extends results by Bending (SDP designs and their automorphism groups, Ph.D. thesis, 1993), and Dempwolff and Neumann (Des. Codes Cryptogr., 57, 373–381, 2010). Finally, we yield new Boolean functions producing two families of a 2-design whose parameters coincide with those of the complement of a point-hyperplane design and a TSDP design, despite being non-isomorphic.

The primary objective of this paper is the construction of optimal codes with self-orthogonality that can be used to construct quantum codes. Recently, Ding and Heng explored subfield codes, which can be viewed as trace codes. In this paper, we focus on investigating self-orthogonal optimal trace codes. First, we provide a novel description of trace codes by choosing suitable defining sets. Second, we determine the parameters of the codes and their trace codes whose defining sets are disjoint union of some affine subspaces in both non-projective cases and projective-cases. This result extends the main findings in (Hu, Li, Zeng, Wang, Tang, IEEE Trans. Inform. Theory, 68(7): 4408-4421, 2022). Third, we compute the parameters of trace codes for MacDonald codes, including the first order Reed-Muller codes and simplex codes as special cases. Finally, we examine their self-orthogonality and distance-optimality to find several classes of self-orthogonal Griesmer codes. Additionally, we resolve a problem proposed by Ding and Heng as a byproduct.

We study the binary linear code families associated with certain types of multivariable functions. We observe that a majority of these codes are not optimal codes nor few weight codes yet. In this paper, we find infinite families offew weight(near-)optimalbinary linear codes from our code families. Furthermore, we produce supportt-designs (t= 2 or 3) which cannot be determined by theAssmus-Mattson Theorem; this is the first time that the result by Tang et al. was successfully used to prove that linear codes holdt-designs. As another application, we find many (near-) optimal quantum codes from the dual codes of our code families using theCSS construction. As a main method, we use themodified shortening method(simply, calledshortening method), which is applied to our code families. Using the results on the weight distributions of our shortened codes, we verify that our codes families supportt-designs (t= 2, 3).We emphasize that some infinite families of few weight optimal binary linear codes have new parameters.

The MacWilliams’ Extension Theorem (MET) with respect to a combinatorial metric states that every isomorphism between linear codes that preserves combinatorial weight can be extended to an isometric extension of the whole space. Pinheiro et al. (2019) proposed the problem of characterizing combinatorial metrics over a finite field with two elements that admit the MET. In this paper, we provide the complete description of such metrics.

Both linear complementary dual (LCD) codes and maximum distance separable (MDS) codes have good algebraic structures, and they have interesting practical applications such as communication systems, data storage, quantum codes, and so on. So far, most of LCD MDS codes have been constructed by employing generalized Reed-Solomon codes. In this paper we construct some classes of new Euclidean LCD MDS codes and Hermitian LCD MDS codes which are not monomially equivalent to Reed-Solomon codes, called LCD MDS codes of non-Reed-Solomon type. Our method is based on the constructions of Beelen et al. (2017) and Roth and Lempel (1989). To the best of our knowledge, this is the first paper on the construction of LCD MDS codes of non-Reed-Solomon type; any LCD MDS code of non-Reed-Solomon type constructed by our method is not monomially equivalent to any LCD code constructed by the method of Carlet et al. (2018).

A linear code is optimal if it has the highest minimum distance of any linear code with a given length and dimension. We construct infinite families of optimal binary linear codes CΔcconstructed from simplicial complexes in F2n, where Δ is a simplicial complex in F2nand Δcthe complement of Δ. We first find an explicit computable criterion for CΔcto be optimal; this criterion is given in terms of the 2-adic valuation of Σsj=12|Ai|-1, where the At's are maximal elements of Δ. Furthermore, we obtain much simpler criteria under various specific conditions on the maximal elements of Δ. In particular, we find that CΔcis a Griesmer code if and only if the maximal elements of Δ are pairwise disjoint and their sizes are all distinct. Specially, when f has exactly two maximal elements, we explicitly determine the weight distribution of CΔc.We present many optimal linear codes constructed by our method, and we emphasize that we obtain at least 32 new optimal linear codes.

Recently, Etzion et al. introduced metrics on F2nbased on directed graphs on n vertices and developed some basic coding theory on directed graph metric spaces. In this paper, we consider the problem of classifying directed graphs, which admit the extended Hamming codes to be a perfect code. We first consider weighted poset metrics as a natural generalization of poset metrics and investigate interrelation between the weighted poset metrics and the directed graph-based metrics. In the next, we classify weighted posets on a set with eight elements and directed graphs on eight vertices, which admit the extended Hamming code H̃3to be a two-perfect code. We also construct some families of such structures for any k ≥ 3, which can be viewed as generalizations of some results presented by Etzion et al. and Hyun and Kim. Those families enable us to construct packing or covering codes of radius 2 under certain maps.

A p-ary function f in n variables is an l-form if f(tu) = tlf (u) for any nonzero t in Zpand u in Zpn. Let n be a positive even integer, p an odd prime, and l an element of {1, 2, . . . , p -1} provided that l ≠ p -1 if p > 3. Let f be a p-ary bent function in n variables of l-form with f (0) = 0 and gcd(l - 1, p - 1) = 1, and let Hl= {tl: t ∈ Zp*}. We denote by Gf,lthe Cayley graph Cay(Zpn, ∪s∈Hlf-1(s)). Our main results are as follows: 1) if there is weakly regular p-ary bent f which is not regular, then l is 2; 2) if l = 2, then f is weakly regular p-ary bent if and only if the Cayley graph G f,l is strongly regular; 3) if l ≠ 2, then f is regular p-ary bent if and only if the Cayley graph Gf,lis strongly regular; 4) Gf,lcan be replaced by Cay(Zpn, f-1(0)\{0}) in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of Gf,l.

Plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics. We find explicit criteria for the construction of p-ary r-plateaued functions with an odd prime p. We point out that 0-plateaued functions are bent functions, and so plateaued functions generalize the notion of bent functions. We first derive an explicit form for the Walsh-Hadamard transform of a p-ary r-plateaued function. We then obtain an upper bound on the degree of p-ary r-plateaued functions, and we classify p-ary (n - 1)-plateaued functions in n variables. We also obtain explicit criteria for the existence of p-ary r-plateaued functions. Accordingly, these results lead to improved bounds on the existence of p-ary bent functions.

We find some necessary conditions for the existence of regular p-ary bent functions (from Znp to Zp), where p is a prime. In more detail, we show that there is no regular p-ary bent function f in n variables with w(Mf) larger than n/2, and for a given nonnegative integer k, there is no regular p-ary bent function f in n variables with w(Mf)=n/2-k ( n+3/2-k, respectively) for an even n ≥ Np,k(an odd n ≥ Np,k, respectively), where Np,kis some positive integer, which is explicitly determined and the w(Mf) of a p-ary function f is some value related to the power of each monomial of f. For the proof of our main results, we use some properties of regular p-ary bent functions, such as the MacWilliams duality, which is proved to hold for regular p-ary bent functions in this paper.

We prove that MDS linear poset-codes satisfy Gilbert-Varshamov bound for their Hamming weights asymptotically. We also construct MDS linear poset-codes on arbitrary poset-metric spaces by using the Dilworth's chain decomposition theorem and results about the Hermite interpolation problem over a finite field. We prove that there exist linear poset-codes with large weights for both poset-metrics and Hamming metrics, as well.

Let P be a poset on $[n]$. We construct a subgroup $\mathcal{G}_P$ of the full poset-isometry group $\mathrm{Iso}_P(F^n_q)$ and present the structure of $\mathcal{G}_P$ as well as its size. We also find poset-metric spaces satisfying $\mathcal{G}_P=\mathrm{Iso}_P(F^n_q)$ by using a characterization of $\mathcal{G}_P$.

We denote Qnthe set of binary words of length n. A partition {C1,C2,¿,Cr} of Qnwith quotient matrix B = (bij)r×ris equitable if for all i and j, any word in d has exactly bij neighbors in Cj. The equitable partitions of Qncan be obtained from completely regular codes. We derive a bound on equitable partitions of Qnthat does not depend on the size of the partition.