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.

The paper provides a thorough investigation of Direct Loss Minimization (DLM), which optimizes the posterior to minimize predictive loss, in sparse Gaussian processes. For the conjugate case, we consider DLM for log-loss and DLM for square loss showing a significant performance improvement in both cases. The application of DLM in non-conjugate cases is more complex because the logarithm of expectation in the log-loss DLM objective is often intractable and simple sampling leads to biased estimates of gradients. The paper makes two technical contributions to address this. First, a new method using product sampling is proposed, which gives unbiased estimates of gradients (uPS) for the objective function. Second, a theoretical analysis of biased Monte Carlo estimates (bMC) shows that stochastic gradient descent converges despite the biased gradients. Experiments demonstrate empirical success of DLM. A comparison of the sampling methods shows that, while uPS is potentially more sample-efficient, bMC provides a better tradeoff in terms of convergence time and computational efficiency.