On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the $L_{\infty }$ norm. We consider algorithms that use standard information ${\mathrm{\Lambda }}^{\mathrm{std}}$ consisting of function values or general linear information ${\mathrm{\Lambda }}^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for ${\mathrm{\Lambda }}^{\mathrm{std}}$ and ${\mathrm{\Lambda }}^{\mathrm{all}}$ under the absolute or normalized error criterion, and show that the power of ${\mathrm{\Lambda }}^{\mathrm{std}}$ is the same as the one of ${\mathrm{\Lambda }}^{\mathrm{all}}$ for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].     

Affine flag varieties and quantum symmetric pairs,II. Multiplication formula

We establish a multiplication formula for a tridiagonal standard basis element in the idempotent version, i.e., the Lusztig form, of the coideal subalgebras of quantum affine ${\mathfrak{gl}}_n$ arising from the geometry of affine partial flag varieties of type C. We apply this formula to obtain the stabilization algebras ${\dot{\mathbf{K}}}_n^{\mathfrak{c}}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$, ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ and ${\dot{\mathbf{K}}}_{\eta }^{??}$, which are idempotented coideal subalgebras of quantum affine ${\mathfrak{gl}}_n$. The symmetry in the formula leads to an isomorphism of the idempotented coideal subalgebras ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ and ${\dot{\mathbf{K}}}_{\mathfrak{n}}^{??}$ with compatible monomial, standard and canonical bases.     

Fq-linear codes over Fq-algebras

Huffman (2013) [12] studied ${\mathbb{F}}_q$-linear codes over ${\mathbb{F}}_{q^m}$ and he proved the MacWilliams identity for these codes with respect to ordinary and Hermitian trace inner products. Let S be a finite commutative ${\mathbb{F}}_q$-algebra. An ${\mathbb{F}}_q$-linear code over S of length n is an ${\mathbb{F}}_q$-submodule of $S^n$. In this paper, we study ${\mathbb{F}}_q$-linear codes over S. We obtain some bounds on minimum distance of these codes, and some large classes of MDR codes are introduced. We generalize the ordinary and Hermitian trace products over ${\mathbb{F}}_q$-algebras and we prove the MacWilliams identity with respect to the generalized form. In particular, we obtain Huffman's results on the MacWilliams identity. Among other results, we give a theory to construct a class of quantum codes and the structure of ${\mathbb{F}}_q$-linear codes over finite commutative graded ${\mathbb{F}}_q$-algebras.     

The cotype zeta function of Zd

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq {\mathbb{Z}}^d$ of index at most $X$ for which ${\mathbb{Z}}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of ${\mathbb{Z}}^d$, a multivariable generalization of the subgroup growth zeta function of ${\mathbb{Z}}^d$.     

Linearity and classification of ZpZp2-linear generalized Hadamard codes

The ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive codes are subgroups of ${\mathbb{Z}}_p^{{\alpha }_1}\times {\mathbb{Z}}_{p^2}^{{\alpha }_2}$, and can be seen as linear codes over ${\mathbb{Z}}_p$ when ${\alpha }_2=0$, ${\mathbb{Z}}_{p^2}$-additive codes when ${\alpha }_1=0$, or ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive codes when $p=2$. A ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}}_p$ which is the Gray map image of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code. Recursive constructions of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive GH codes of type $({\alpha }_1,{\alpha }_2;t_1,t_2)$ with $t_1,t_2\ge 1$ are known. In this paper, we generalize some known results for ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with $p=2$ to any $p\ge 3$ prime when ${\alpha }_1\ne 0$, and then we compare them with the ones obtained when ${\alpha }_1=0$. First, we show for which types the corresponding ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes are nonlinear over ${\mathbb{Z}}_p$. Then, for these codes, we compute the kernel and its dimension, which allow us to classify them completely. Moreover, by computing the rank of some of these codes, we show that, unlike ${\mathbb{Z}}_4$-linear Hadamard codes, the ${\mathbb{Z}}_{p^2}$-linear GH codes are not included in the family of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with ${\alpha }_1\ne 0$ when $p\ge 3$ prime. Indeed, there are some families with infinite nonlinear ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes, where the codes are not equivalent to any ${\mathbb{Z}}_{p^s}$-linear GH code with $s\ge 2$.