In this paper, we present a new construction of asymmetric quantum codes (AQCs) by combining classical concatenated codes (CCs) with tensor product codes (TPCs), called asymmetric quantum concatenated and tensor product codes (AQCTPCs) which have the following three advantages. First, only the outer codes in AQCTPCs need to satisfy the orthogonal constraint in quantum codes, and any classical linear code can be used for the inner, which makes AQCTPCs very easy to construct. Second, most AQCTPCs are highly degenerate, which means they can correct many more errors than their classical TPC counterparts. Consequently, we construct several families of AQCs with better parameters than known results in the literature. Third, AQCTPCs can be efficiently decoded although they are degenerate, provided that the inner and outer codes are efficiently decodable. In particular, we significantly reduce the inner decoding complexity of TPCs from $\Omega (n_{2}a^{n_{1}})(a>1)$ to $O(n_{2})$ by considering error degeneracy, where $n_{1}$ and $n_{2}$ are the block length of the inner code and the outer code, respectively. Furthermore, we generalize our concatenation scheme by using the generalized CCs and TPCs correspondingly.

In this paper, we utilize a concatenation scheme to construct new families of quantum error correction codes achieving the quantum Gilbert-Varshamov (GV) bound asymptotically. Weconcatenate alternant codes with any linear code achievingthe classical GV bound to construct Calderbank-Shor-Steane (CSS) codes. We show that the concatenated code can achieve the quantum GV bound asymptotically and can approach the Hashing bound for asymmetric Pauli channels. By combing Steane’s enlargement construction of CSS codes, we derive a family of enlarged stabilizer codes achieving the quantum GV bound for enlarged CSS codes asymptotically. Asapplications, we derive two families of fast encodable and decodable CSS codes with parameters $\mathscr {Q}_{1}=[[N,\Omega (\sqrt {N}),\Omega (\sqrt {N})]]$ , and $\mathscr {Q}_{2}=[[N,\Omega (N/\log N),\Omega (N/\log N)/\Omega (\log N)]]$ . We show that $\mathscr {Q}_{1}$ can be encoded very efficiently by circuits of size $O(N)$ and depth $O(\sqrt {N})$ . For an input error syndrome, $\mathscr {Q}_{1}$ can correct any adversarial error of weight up to half the minimum distance bound in $O(N)$ time. $\mathscr {Q}_{1}$ can also be decoded in parallel in $O(\sqrt {N})$ time by using $O(\sqrt {N})$ classical processors. For an input error syndrome, we proved that $\mathscr {Q}_{2}$ can correct a linear number of ${X}$ -errors with high probability and an almost linear number of ${Z}$ -errors in $O(N)$ time. Moreover, $\mathscr {Q}_{2}$ can be decoded in parallel in $O(\log (N))$ time by using $O(N)$ classical processors.

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Code-based masking is one of the most important countermeasures to prevent side-channel attacks. Dual distance and kissing number of linear codes are critical metrics indicating the side-channel resistance of code-based masking. However, calculating the value of these indicators depends on the choice of irreducible polynomials of the corresponding finite field. Through simulation experiments, we discovered that different irreducible polynomials significantly impact the side-channel resistance. Our findings suggest that the optimal linear codes derived from a single irreducible polynomial may not be globally optimal. By going through all irreducible polynomials over GF(256), we identify the optimal linear codes for 2-Share inner product masking (IPM) and (3,1)-Shamir’s secret sharing (SSS) based masking in this field. Furthermore, we explore the application of higher-order masking defenses to the lightweight encryption scheme PRESENT. Additionally, we demonstrate through comprehensive simulation experiments that 3-share IPM provides substantial security advantages over 2-share IPM. We also extend our analysis to GF(32) and GF(64) to validate the generalizability of our approach.

In most practical quantum mechanical systems, quantum noise due to decoherence is highly biased towards dephasing. The quantum state suffers from phase flip noise much more seriously than from the bit flip noise. In this work, we construct new families of asymmetric quantum concatenated codes (AQCCs) to deal with such biased quantum noise. Our construction is based on a novel concatenation scheme for constructing AQCCs with large asymmetries, in which classical tensor product codes and concatenated codes are utilized to correct phase flip noise and bit flip noise, respectively. We generalize the original concatenation scheme to a more general case for better correcting degenerate errors. Moreover, we focus on constructing nonbinary AQCCs that are highly degenerate. Compared to previous literatures, AQCCs constructed in this paper show much better parameter performance than existed ones. Furthermore, we design the specific encoding circuit of the AQCCs. It is shown that our codes can be encoded more efficiently than standard quantum codes.