Folded Reed-Solomon codes, introduced by Guruswami and Rudra in 2007, have been shown to achieve the information-theoretically best possible trade-off between the rate of a code and the error-correction radius. In 2024, Bergamaschi, Golowich and Gunn extended this framework by constructing folded quantum Reed-Solomon codes (CSS codes obtained by folding) demonstrating that these codes tolerate errors up to the quantum Singleton bound. In this paper, we construct folded quantum Hermitian codes using the CSS framework and show that these codes are also list-decodable, tolerating errors up to the quantum Singleton bound. Compared to Reed-Solomon codes, Hermitian codes admit comparable lengths over smaller alphabets, enabling more efficient implementations.
Quantum list recovery (QLR) and quantum list decoding (QLD) seek short lists of logically distinct Pauli corrections consistent with a syndrome and prescribed error constraints. For CSS codes, two issues arise: treating the
X- and
Z-sectors separately can multiply their output list sizes, while distinct classical candidates can collapse after stabilizer quotienting.
We study combinatorial upper and lower bounds for balanced folded quantum Reed--Solomon (FQRS) codes and balanced random CSS codes, both studied by Bergamaschi, Golowich, and Gunn (STOC 24). Let
R∈(0,1) be the quantum rate and
R1=(1+R)/2 the common component rate. As the radius
ρ=(1−R)/2−γ approaches the quantum Singleton bound, we have, deterministically for FQRS codes and w.h.p. for balanced random CSS codes,
LQLR⋆=ℓΘ(R1/γ),LQLD⋆=Θ(γ1−R)(γ↓0). The asymptotically exact QLD radius tradeoff is
ρL⋆=L+1L21−R. These conclusions extend to the average-radius setting.
For achievability, building on the work of Brakensiek, Chen, Dhar, and Zhang (STOC 2026), we establish a pairing lemma for joint
X/Z candidate lists that preserves the one-sector coefficient and avoids a product loss in list size. For the QLD converse, we prove a quantum generalized Singleton bound based on the classical projection-and-patching argument with stabilizer distinctness. For the QLR lower bounds, we adapt the folded Reed--Solomon construction of Chen and Zhang (STOC 2025) so that the candidates remain stabilizer distinct. For random CSS codes, we show classical bad lists survive the stabilizer quotient with high probability.
Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.
Hermitian hulls of linear codes are interesting for theoretical and practical reasons alike. In terms of recent application, linear codes whose hulls meet certain conditions have been utilized as ingredients to construct entanglement-assisted quantum error correcting codes. This family of quantum codes is often seen as a generalization of quantum stabilizer codes. Theoretically, compared with the Euclidean setup, the Hermitian case is much harder to deal with. Hermitian hulls of MDS linear codes with low dimensions have been explored, mostly from generalized Reed-Solomon codes. Characterizing Hermitian hulls which themselves are MDS appears to be more involved and has not been extensively studied. This paper introduces some tools to study linear codes whose Hermitian hulls are MDS. Using the tools, we then propose explicit constructions of such codes. We consider Hermitian hulls of both Reed-Solomon and non Reed-Solomon types of linear MDS codes. We demonstrate that, given the same Hermitian hull dimensions, the codes from our constructions have dimensions which are larger than those in the literature.
Quantum error-correcting codes are essential for protecting quantum information against decoherence. This work investigates the construction of quantum CSS codes from classical self-orthogonal binary linear codes derived from extended quadratic residue codes. By exploiting the large automorphism groups of these codes, the minimum-weight codewords are partitioned according to the action of maximal subgroups of the automorphism group. Codes generated from these partitions are shown to be self-orthogonal and to retain large automorphism groups, allowing the construction of associated quantum CSS codes with good parameters. The resulting codes include new quantum codes with lengths ranging from 16 to 48 and minimum distances up to 11. Their automorphism groups and permutation-decoding properties are also analyzed, including the number of permutations required for complete decoding. The results provide new families of quantum CSS codes combining good error-correcting parameters, large automorphism groups, and efficient permutation decoding.
Fault tolerant quantum computation is a critical step in the development of practical quantum computers. Unfortunately, not every quantum error correcting code can be used for fault tolerant computation. Rengaswamy et. al. define CSS-T codes, which are CSS codes that admit the transversal application of the T gate, which is a key step in achieving fault tolerant computation. They then present a family of quantum Reed-Muller fault tolerant codes. Their family of codes admits a transversal T gate, but the asymptotic rate of the family is zero. We build on their work by reframing their CSS-T conditions using the concept of self-orthogonality. Using this framework, we define an alternative family of quantum Reed-Muller fault tolerant codes. Like the quantum Reed-Muller family found by Rengaswamy et. al., our family admits a transversal T gate, but also has a nonvanishing asymptotic rate.\nWe prove three key results in our search for a Reed-Muller CSS-T family with a nonvanishing rate. First, we show an equivalence between a code containing a self-dual subcode and the dual of that code being self-orthogonal. This allows us to more easily determine if a pair of codes define a CSS-T code. Next, we show that if C1 and C2 are both Reed-Muller codes that form a CSS-T code, C1 must be self-orthogonal. This limits the rate of any family that is constructed solely from Reed-Muller codes. Lastly, we define a family of CSS-T codes by choosing C1 = RM(r, 2r + 1) and C2 = RM(0, 2r + 1) for some nonnegative integer r. We show that this family has an asymptotic rate of 1/2, and show that it is the only possible CSS-T family constructed only from Reed-Muller codes where C1 is self dual.