bibkey: berardini2025csst
authors: Elena Berardini; Reza Dastbasteh; Josu Etxezarreta Martinez; Shreyas Jain; Olatz Sanz Larrarte
year: 2025
title: "Asymptotically Good CSS-T Codes and a New Construction of Triorthogonal Codes"
doi: 10.1109/JSAIT.2025.3582156
url: https://arxiv.org/abs/2412.08586v2
claim: "From a binary CSS code (C1, C2) and a linear map φ on C1 satisfying a triple-product parity condition, the paper builds the CSS-T code (C1^{Nφ}, C2^{Nφ}) with C^{Nφ} = {(x, φ(x))}, proves that φ = I gives a [[2n, k, ≥ d]] CSS-T code, and asks for maps φ giving better parameters than the identity or a permutation."
strata_touched:
  - D5/S3/Quantum/Codes/CssTDoublingMapImprovement
license: citation-only
triage: anchor

Asymptotically Good CSS-T Codes and a New Construction of Triorthogonal Codes

E. Berardini, R. Dastbasteh, J. Etxezarreta Martinez, S. Jain and O. Sanz Larrarte, arXiv:2412.08586v2 (v1 2024-12-11); IEEE J. Sel. Areas Inf. Theory 6 (2025) 189–198. Quotations are from the v2 TeX source.

CSS parameters (Theorem 1):

Let $C_2,C_1\subseteq {\mathbb{F}}_2^n$ be two linear codes such that $C_2\subseteq C_1$. Then, there exists an $[[n,k,d]]$ quantum code, called CSS and denoted by $(C_1,C_2{)}_{\mathrm{CSS}}$, where $k=\dim (C_1)-\dim (C_2)$ and $d=\min \{\widetilde{(}{C}_1\setminus C_2),\widetilde{(}{C}_2^{\perp }\setminus C_1^{\perp })\}$.

The construction (Definition 3):

Let $C\subseteq {\text{F}}_2^n$ be a linear code. Let $\varphi :C\to {\text{F}}_2^n$ be a map. We define $$C^{N_{\varphi }}\text{defeq}\left\{(x,\varphi (x))\mid x\in C\right\}.$$

The admissibility condition (Proposition 1):

Then $\left(C_1^{N_{\varphi }},C_2^{N_{\varphi }}\right)$ forms a CSS-T code if and only if $\varphi :C_1\to {\text{F}}_2^n$ is a linear map such that for each $x,y\in C_1$ and $z\in C_2$ we have $$\widetilde{(}x\star y\star z)+\widetilde{(}\varphi (x)\star \varphi (y)\star \varphi (z))\equiv 0\mathrm{mod}2.$$

The open question (Remark 1):

Finding maps $\varphi$ giving CSS-T codes with better parameters than the choice $\varphi =I$ or a permutation is an interesting open question.

Verified locator

• URL: https://arxiv.org/abs/2412.08586v2 (source CSST-JSAIT-final.tex, md5 0a12219774df7122726d3bd6783a16f0):
  • Theorem 1, l. 199–201;
  • Definition 3, l. 275–279;
  • Proposition 1, l. 282–285;
  • Proposition 3, l. 306–321;
  • Remark 1, l. 330–332;
  • the conclusion, l. 695.
• DOI: https://doi.org/10.1109/JSAIT.2025.3582156.