An official website of the United States government
Abstract CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes$$(C_1, C_2)$$ such that$$C_1$$ contains$$C_2$$ ,$$C_2$$ is even, and the shortening of the dual of$$C_1$$ with respect to the support of each codeword of$$C_2$$ is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes$$(C_1, C_2)$$ is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.
more »
« less
Torsion phenomena for zero-cycles on a product of curves over a number field
Abstract For a smooth projective varietyXover an algebraic number fieldka conjecture of Bloch and Beilinson predicts that the kernel of the Albanese map ofXis a torsion group. In this article we consider a product$$X=C_1\times \cdots \times C_d$$ of smooth projective curves and show that if the conjecture is true for any subproduct of two curves, then it is true forX. For a product$$X=C_1\times C_2$$ of two curves over$$\mathbb {Q} $$ with positive genus we construct many nontrivial examples that satisfy the weaker property that the image of the natural map$$J_1(\mathbb {Q})\otimes J_2(\mathbb {Q})\xrightarrow {\varepsilon }{{\,\textrm{CH}\,}}_0(C_1\times C_2)$$ is finite, where$$J_i$$ is the Jacobian variety of$$C_i$$ . Our constructions include many new examples of non-isogenous pairs of elliptic curves$$E_1, E_2$$ with positive rank, including the first known examples of rank greater than 1. Combining these constructions with our previous result, we obtain infinitely many nontrivial products$$X=C_1\times \cdots \times C_d$$ for which the analogous map$$\varepsilon $$ has finite image.
more »
« less
- Award ID(s):
- 2302196
- PAR ID:
- 10497259
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Research in Number Theory
- Volume:
- 10
- Issue:
- 2
- ISSN:
- 2522-0160
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ for$$j=1,\dots ,m$$ with coefficients$$a_{j,i}\in \mathbb {F}_p$$ . Suppose that$$k\ge 3m$$ , that$$a_{j,1}+\dots +a_{j,k}=0$$ for$$j=1,\dots ,m$$ and that every$$m\times m$$ minor of the$$m\times k$$ matrix$$(a_{j,i})_{j,i}$$ is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$ of size$$|A|> C\cdot \Gamma ^n$$ contains a solution$$(x_1,\dots ,x_k)\in A^k$$ to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$ are all distinct. Here,Cand$$\Gamma $$ are constants only depending onp,mandksuch that$$\Gamma . The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$ in the solution$$(x_1,\dots ,x_k)\in A^k$$ to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.more » « less
-
Abstract Let$$\mathbb {F}_q^d$$ be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ and a fixed nonzero$$t\in \mathbb {F}_q$$ , let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ , where$$h_y:E\rightarrow \{0,1\}$$ is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ . Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ that if$$|E|\ge Cq^{\frac{11}{4}}$$ andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ isdwhenever$$E\subseteq \mathbb {F}_q^d$$ with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ .more » « less
-
Abstract Schinzel and Wójcik have shown that for every$$\alpha ,\beta \in \mathbb {Q}^{\times }\hspace{0.55542pt}{\setminus }\hspace{1.111pt}\{\pm 1\}$$ , there are infinitely many primespwhere$$v_p(\alpha )=v_p(\beta )=0$$ and where$$\alpha $$ and$$\beta $$ generate the same multiplicative group modp. We prove a weaker result in the same direction for algebraic numbers$$\alpha , \beta $$ . Let$$\alpha , \beta \in \overline{\mathbb {Q}} ^{\times }$$ , and suppose$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\alpha )|\ne 1$$ and$$|N_{\mathbb {Q}(\alpha ,\beta )/\mathbb {Q}}(\beta )|\ne 1$$ . Then for some positive integer$$C = C(\alpha ,\beta )$$ , there are infinitely many prime idealsPof Equation missing<#comment/>where$$v_P(\alpha )=v_P(\beta )=0$$ and where the group$$\langle \beta \bmod {P}\rangle $$ is a subgroup of$$\langle \alpha \bmod {P}\rangle $$ with$$[\langle \alpha \bmod {P}\rangle \,{:}\, \langle \beta \bmod {P}\rangle ]$$ dividingC. A key component of the proof is a theorem of Corvaja and Zannier bounding the greatest common divisor of shiftedS-units.more » « less
