Robust Syndrome Extraction via BCH Encoding

Quantum data-syndrome (QDS) codes are a class of quantum error-correcting codes that protect against errors both on the data qubits and on the syndrome itself via redundant measurement of stabilizer group elements. One way to define a QDS code is to choose a syndrome measurement code, a classical block code that encodes the syndrome of the underlying quantum code by defining additional stabilizer measurements. We propose the use of primitive narrow-sense BCH codes as syndrome measurement codes. We show that these codes asymptotically require $O(t\log\ell)$ extra measurements, where $\ell$ is the number of stabilizer generators of the quantum code and $t$ is the number of errors corrected by the BCH code. Previously, the best known general method of constructing QDS codes out of quantum codes requires $O(t^3\log\ell)$ extra measurements. As the number of additional syndrome measurements is a reasonable metric for the amount of additional time a general QDS code requires, we conclude that our construction protects against the same number of syndrome errors with significantly less time overhead

A class of narrow-sense BCH codes over Fq\mathbb{F}_q of length qm−12\frac{q^m-1}{2}

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length $\frac{q^m-1}{2}$ over $\mathbb{F}_q$ with special trace representation, where $q$ is an odd prime power. With the help of the inner distributions of some subsets of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. From determining some cyclotomic coset leaders $\delta_i$ of cyclotomic cosets modulo $\frac{q^m-1}{2}$, we prove that narrow-sense BCH codes of length $\frac{q^m-1}{2}$ with designed distance $\delta_i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor+i}-1}{2}$ have the corresponding trace representation, and have the minimal distance $d=\delta_i$ and the Bose distance $d_B=\delta_i$, where $1\leq i\leq \lfloor \frac{m+3}{4} \rfloor$

Algebraic techniques in designing quantum synchronizable codes

Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the previously known general framework for designing quantum synchronizable codes through more extensive use of the theory of finite fields. This makes it possible to widen the range of tolerable magnitude of block synchronization errors while giving mathematical insight into the algebraic mechanism of synchronization recovery. Also given are families of quantum synchronizable codes based on punctured Reed-Muller codes and their ambient spaces.Comment: 9 pages, no figures. The framework presented in this article supersedes the one given in arXiv:1206.0260 by the first autho