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.

Bent partitions ofV(p)nplay an important role in constructing (vectorial) bent functions, partial difference sets, and association schemes, whereV(p)ndenotes ann-dimensional vector space over the finite field Fp,nis an even positive integer, and p is a prime. It is a challenging open problem whether the depth of any bent partition ofV(p)nis always a power ofp. Notably, the depths of all currently known bent partitions ofV(p)nare powers ofp. In this paper, we prove that for a bent partition Γ ofV(p)nfor which all thep-ary bent functions generated by Γ are regular or all are weakly regular but not regular, the depth of Γ must be a power ofp. We present new constructions of bent partitions that (do not) correspond to vectorial dual-bent functions. In particular, a new construction of vectorial dual-bent functions is provided. Additionally, for general bent partitions ofV(2)n, we establish a characterization in terms of Hadamard matrices.

Self-orthogonal codes are a significant class of linear codes in coding theory and have attracted a lot of attention. In [20], [26],p-ary self-orthogonal codes were constructed by usingp-ary weakly regular bent functions, wherepis an odd prime. In [42], two classes of non-degenerate quadratic forms were used to construct q-ary self-orthogonal codes, whereqis a power of a prime. In this paper, we construct new families ofq-ary self-orthogonal codes using vectorial dual-bent functions. Some classes of at least almost optimal linear codes are obtained from the dual codes of the constructed self-orthogonal codes. In some cases, we completely determine the weight distributions of the constructed self-orthogonal codes. From the view of vectorial dual-bent functions, we illustrate that the works on constructing self-orthogonal codes fromp-ary weakly regular bent functions [20], [26] and non-degenerate quadratic forms withqbeing odd [42] can be obtained by our results. We partially answer an open problem on determining the weight distribution of a class of self-orthogonal codes given in [42]. As applications, we construct new infinite families of at least almost optimalq-ary linear complementary dual codes (for short, LCD codes) and quantum codes.

Self-orthogonal codes have received great attention due to their important applications in quantum codes, LCD codes and lattices. Recently, several families of self-orthogonal codes containing the all-1 vector were constructed by augmentation technique. In this paper, utilizing plateaued functions, we construct some classes of linear codes which do not contain the all-1 vector. We also investigate their punctured codes. The weight distributions of the constructed codes are explicitly determined. Under certain conditions, these codes are proved to be self-orthogonal. Furthermore, some classes of optimal linear codes are obtained from their duals. Using the self-orthogonal punctured codes, we also construct several new families of at least almost optimal quantum codes and optimal LCD codes.

Vectorial dual-bent functions have recently attracted some researchers’ interest as they play a significant role in constructing partial difference sets, association schemes, bent partitions, and linear codes. In this paper, we further study vectorial dual-bent functions$F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$, where$2\leq m \leq \frac {n}{2}$, and$V_{n}^{(p)}$denotes an n-dimensional vector space over the prime field$\mathbb {F}_{p}$. For certain vectorial dual-bent functions (called vectorial dual-bent functions with Condition A), we present a more concise characterization in terms of partial difference sets than the one given in Wang et al. (2023), and give new characterizations in terms of amorphic association schemes, linear codes, and generalized Hadamard matrices, respectively. When$p=2$, we characterize vectorial dual-bent functions with Condition A in terms of bent partitions. Through the relationship between vectorial dual-bent functions and bent partitions, new characterizations of certain bent partitions in terms of amorphic association schemes, linear codes, and generalized Hadamard matrices are obtained. For a vectorial dual-bent function$F: V_{n}^{(p)}\rightarrow V_{m}^{(p)}$with$F(0)=0, F(x)=F(-x)$, where$2\leq m \leq \frac {n}{2}$, we give a necessary and sufficient condition under which the preimage set partition of F induces an association scheme. By using two classes of vectorial dual-bent functions, more association schemes are obtained.

It is well-known that MacWilliams identities play a significant role in coding theory. In [5]-[7], MacWilliams-like identities for p-ary bent functions $f:\mathbb{F}_p^n \to {\mathbb{F}_p}$ were given, where p is a prime. The aim of this paper is to investigate MacWilliams-like identities for vectorial bent functions. We give MacWillaims-like identities for certain vectorial bent functions $F:\mathbb{F}_q^t \to {\mathbb{F}_q}$, where q is a power of a prime p. We illustrate that when q = p, the MacWilliams-like identities for weakly regular p-ary bent functions can be obtained by our results. Based on the obtained MacWilliams-like identities, we give some nonexistence results on vectorial bent functions.

Bent partitions of$V_{n}^{(p)}$are quite powerful in constructing bent functions, vectorial bent functions and generalized bent functions, where$V_{n}^{(p)}$is an$n$-dimensional vector space over$\mathbb {F}_{p}$,$n$is an even positive integer and$p$is a prime. The classical examples of bent partitions are obtained from (partial) spreads. In Anbar and Meidl (2022) and Meidl and Pirsic (2021), two classes of bent partitions which are not obtained from (partial) spreads were presented. In Anbar et al. (2023), more bent partitions$\Gamma _{1}, \Gamma _{2}, \Gamma _{1}^{\bullet }, \Gamma _{2}^{\bullet }, \Theta _{1}, \Theta _{2}$were presented from (pre)semifields, including the bent partitions given in Anbar and Meidl (2022) and Meidl and Pirsic (2021). In this paper, we investigate the relations between bent partitions and vectorial dual-bent functions. For any prime$p$, we show that one can generate certain bent partitions (called bent partitions satisfying Condition$\mathcal {C}$) from certain vectorial dual-bent functions (called vectorial dual-bent functions satisfying Condition A). In particular, when$p$is an odd prime, we show that bent partitions satisfying Condition$\mathcal {C}$one-to-one correspond to vectorial dual-bent functions satisfying Condition A. We give an alternative proof that$\Gamma _{1}, \Gamma _{2}, \Gamma _{1}^{\bullet }, \Gamma _{2}^{\bullet }, \Theta _{1}, \Theta _{2}$are bent partitions in terms of vectorial dual-bent functions. We present a secondary construction of vectorial dual-bent functions, which can be used to generate more bent partitions. We show that any weakly regular ternary bent function$f: V_{n}^{(3)}\rightarrow \mathbb {F}_{3}$($n$is even) of 2-form can generate a bent partition. When such$f$is weakly regular but not regular, the generated bent partition from$f$is not coming from a normal bent partition, which answers an open problem proposed in Anbar and Meidl (2022). We give a sufficient condition on constructing partial difference sets from bent partitions, and when$p$is an odd prime, we provide a characterization of bent partitions satisfying Condition$\mathcal {C}$in terms of partial difference sets.

A function $f:\mathbb{F}_2^n \to {\mathbb{F}_2}$ is called a 5-valued spectrum function if the Walsh transform Wftakes the values $0, \pm {2^{\frac{{n + {s_1}}}{2}}}, \pm {2^{\frac{{n + {s_2}}}{2}}}$ for some different non-negative integers si,i = 1,2 with n + sieven. In [IEEE Transactions on Information Theory, 67 (2), 2021], by spectral method, Hodžić et al. characterized the so-called basic 5-valued spectrum functions f whose duals $f_{[i]}^{\ast}(i = 1,2)$ are totally disjoint spectra functions. For a special case that the corresponding functions $\overline {f_{[i]}^{\ast}} (i = 1,2)$ are basic plateaued functions, Hodžić et al. gave a construction of basic 5-valued spectrum functions. They left open problems to provide constructions of 5-valued spectrum functions f such that f are basic and the corresponding functions $\overline {f_{[i]}^{\ast}} (i = 1,2)$ are non-basic plateaued functions, or f are non-basic. In this paper, we provide three new constructions of 5-valued spectrum functions f with totally disjoint spectra duals $f_{[i]}^{\ast}(i = 1,2)$, including the case that f are basic and the corresponding functions $\overline {f_{[i]}^{\ast}} (i = 1,2)$ are non-basic plateaued functions, and the case that f are non-basic, which provide answers to the problems proposed by Hodžić et al.. Since non-basic 5-valued spectrum functions are EA-inequivalent to basic ones, two constructions in this paper can produce 5-valued spectrum functions which are EA-inequivalent to ones in [IEEE Transactions on Information Theory, 67 (2), 2021].

In this paper, we study the duals of generalized bent functions$f: V_{n}\rightarrow \mathbb {Z}_{p^{k}}$, where$V_{n}$is an$n$-dimensional vector space over$\mathbb {F}_{p}$and$p$is an odd prime,$k$is a positive integer. It is known that weakly regular generalized bent functions always appear in pairs since the dual of a weakly regular generalized bent function is also a weakly regular generalized bent function. The duals of non-weakly regular generalized bent functions can be generalized bent or not generalized bent. By generalizing the construction of Çeşmelioğluet al., 2016, we provide an explicit construction of generalized bent functions whose duals can be generalized bent or not generalized bent. We show that the well-known direct sum construction and the generalized indirect sum construction given in Wang and Fu, 2021. can provide secondary constructions of generalized bent functions whose duals can be generalized bent or not generalized bent. By using the knowledge on ideal decomposition in cyclotomic fields, we prove that$f^{**}(x)=f(-x)$if$f$is a generalized bent function and its dual$f^{*}$is also a generalized bent function. For any non-weakly regular generalized bent function$f$which satisfies that$f(x)=f(-x)$and its dual$f^{*}$is generalized bent, we give a property and as a consequence, we prove that there is no self-dual generalized bent function$f: V_{n}\rightarrow \mathbb {Z}_{p^{k}}$if$p\equiv 3 ~(mod ~4)$and$n$is odd. When$p \equiv 1 ~(mod ~4)$or$p\equiv 3 ~(mod ~4)$and$n$is even, we give a secondary construction of self-dual generalized bent functions. In the end, by the decomposition of generalized bent functions, we characterize the relations between the generalized bentness of the dual of a generalized bent function$f$and the bentness of the duals of bent functions associated with the generalized bent function$f$, as well as the relations of self-duality between a generalized bent function$f$and bent functions associated with the generalized bent function$f$.