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arXiv:1611.06673 · IEEE Transactions on Information Theory

On the List-Decodability of Random Self-Orthogonal Codes

Lingfei Jin, Chaoping Xing, Xiande Zhang

Published Oct 3, 2014• 4 citations
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Abstract

Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.

DOI
10.1109/tit.2014.2361333

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Quantum self-dual codes and symmetric matrices

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This paper has been withdrawn since a Gilbert-Varshamov bound for general quantum codes has already appeared in Ekert and Macchiavello, Prys. Rev. Lett. 77, p. 2585, and a Gilbert-Varshamov bound for stabilizer codes connected with orthogonal geometry, or equivalently, with symmetric matrices as in this paper, has been proved by Calredbank, Rains, Shor and Sloane, Phys. Rev. Lett. 78, p. 405. I would like to thank Robert Calderbank for pointing out these references to me.

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It is well known that quantum codes can be constructed through classical symplectic self-orthogonal codes. In this paper, we give a kind of Gilbert-Varshamov bound for symplectic self-orthogonal codes first and then obtain the Gilbert-Varshamov bound for quantum codes. The idea of obtaining the Gilbert-Varshamov bound for symplectic self-orthogonal codes follows from counting arguments.

arXiv:1308.3578•IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2011•16 citations

Improvement of the Gilbert-Varshamov Bound for Linear Codes and Quantum Codes

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The Gilbert--Varshamov (GV) bound is a central benchmark in coding theory, establishing existential guarantees for error-correcting codes and serving as a baseline for both Hamming and quantum fault-tolerant information processing. Despite decades of effort, improving the GV bound is notoriously difficult, and known improvements often rely on technically heavy arguments and do not extend naturally to the quantum setting due to additional self-orthogonality constraints. In this work we develop a concise probabilistic method that yields an improvement over the classical GV bound for $q$-ary linear codes. For relative distance $δ=d/n<1-1/q$, we show that an $[n,k,d]_q$ linear code exists whenever $\frac{q^{k}-1}{q-1}\;<\;\frac{c_δ\sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}$, for positive constant $c_δ$ depending only on $δ$, where $\mathrm{Vol}_q(n,d-1)$ denotes the volume of a $q$-ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For $δ<1-1/q^2$, we obtain an improved quantum GV bound: there exists a $q$-ary quantum code $[[n,\,n-k,\,d]]$ provided that $\frac{q^{2n-k}-1}{q-1}<\frac{c_δ\sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i}$. In particular, our result improves the standard quantum GV bound by an $Ω(\sqrt{n})$ multiplicative factor.

arXiv:2601.18590•Jan 26, 2026•0 citations

Quantum Generalized Reed-Solomon codes concatenated with random rate one inner stabilizer codes asymptotically attain the Quantum Gilbert-Varshamov bound

Yingkai Ouyang

AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets the Quantum Singleton bound. Fixing the outer code rate and letting n grow,the concatenated stabilizer code has a distance that almost surely attains the QGVB.This partially generalizes Thommesen’s result, where heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1

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QUANTUM SELF-DUAL CODES AND SYMMETRIC MATRICES

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•Dec 1, 2000•0 citations

Random Quantum LDPC Codes Approaching the Gilbert-Varshamov Bound

Tushant Mittal, Shashank Srivastava, Madhur Tulsiani +1 more

We show that for any $R,ε>0$ and any prime $p\geq 2$, there exists an infinite family of $p$-ary quantum low-density parity-check (QLDPC) codes, rate $R$, checks of weight $O_ε(1)$, and normalized distance at least $δ_{\mathrm{GV}}(p,R)-ε$. Here, $δ_{\mathrm{GV}}(p,R)$ denotes the quantum Gilbert-Varshamov (GV) bound for $p$-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.

cs.ITCross-listedarXiv:2610.02648•Oct 5, 2026•0 citations
Guruswami et al. showed that the list-decodability of random linear codes is as good as that of general random codes. In this paper, we further strengthen the result by showing that the list-decodability of random Euclidean self-orthogonal codes is as good as that of general random codes as well, i.e., achieves the classical Gilbert-Varshamov bound. In particular, we show that, for any fixed finite field Fq, error fraction δ ∈ (0,1 - 1/q) satisfying 1 - Hq(δ) ≤ 1/2, and small ε > 0, with high probability a random Euclidean self-orthogonal code over Fqof rate 1 - Hq(δ) - ε is (δ, O(1/ε))-list-decodable. This generalizes the result of linear codes to Euclidean self-orthogonal codes. In addition, we extend the result to list decoding symplectic dual-containing codes by showing that the list-decodability of random symplectic dual-containing codes achieves the quantum Gilbert-Varshamov bound as well. This implies that list-decodability of quantum stabilizer codes can achieve the quantum Gilbert-Varshamov bound. The counting argument on self-orthogonal codes is an important ingredient to prove our result.
The Gilbert--Varshamov (GV) bound is a central benchmark in coding theory, establishing existential guarantees for error-correcting codes and serving as a baseline for both Hamming and quantum fault-tolerant information processing. Despite decades of effort, improving the GV bound is notoriously difficult, and known improvements often rely on technically heavy arguments and do not extend naturally to the quantum setting due to additional self-orthogonality constraints. In this work we develop a concise probabilistic method that yields an improvement over the classical GV bound for qqq-ary linear codes. For relative distance δ=d/n<1−1/qδ=d/n<1-1/qδ=d/n<1−1/q, we show that an [n,k,d]q[n,k,d]_q[n,k,d]q​ linear code exists whenever qk−1q−1  <  cδn qnVolq(n,d−1)\frac{q^{k}-1}{q-1}\;<\;\frac{c_δ\sqrt{n}\, q^{n}}{\mathrm{Vol}_q(n,d-1)}q−1qk−1​<Volq​(n,d−1)cδ​n​qn​, for positive constant cδc_δcδ​ depending only on δδδ, where Volq(n,d−1)\mathrm{Vol}_q(n,d-1)Volq​(n,d−1) denotes the volume of a qqq-ary Hamming ball. We further adapt this approach to the quantum setting by analyzing symplectic self-orthogonal structures. For δ<1−1/q2δ<1-1/q^2δ<1−1/q2, we obtain an improved quantum GV bound: there exists a qqq-ary quantum code [[n, n−k, d]][[n,\,n-k,\,d]][[n,n−k,d]] provided that q2n−k−1q−1<cδn⋅q2n∑i=0d−1(ni)(q2−1)i\frac{q^{2n-k}-1}{q-1}<\frac{c_δ\sqrt{n}\cdot q^{2n}}{\sum_{i=0}^{d-1}\binom{n}{i}(q^2-1)^i}q−1q2n−k−1​<∑i=0d−1​(in​)(q2−1)icδ​n​⋅q2n​. In particular, our result improves the standard quantum GV bound by an Ω(n)Ω(\sqrt{n})Ω(n​) multiplicative factor.
AbstractA good quantum code corrects for a linear number of errors. It has been shownthat random codes attains the QGVB with a relative distance H −1q 2 ((1−R)/2) wherethe code is q-ary and R is the rate of the code (number of encoded systems / blocklength). However,randomcodeshavelittlestructure. Inthispaper,westudyafamilyofconcatenatedq-arystabilizercodes. Eachoftheinnercodesisarandomrate1q-arystabilizer code of block length n. The outer code is a quantum MDS code with blocklengthq n andalphabetsizeq n ,anarbitraryrateR≤1,anddistanceN(1−R)/2+1thatmeets the Quantum Singleton bound. Fixing the outer code rate and letting n grow,the concatenated stabilizer code has a distance that almost surely attains the QGVB.This partially generalizes Thommesen’s result, where heshowed thatthedistanceof aconcatenated code with a Reed-Solomon outer code and random inner linear codes ofarbitraryrateattainstheGilbert-Varshamovboundalmostsurely. 1 Introduction A family of q-ary quantum codes of increasing block length is defined to be good if the ratioof its distance to its block length approaches a non-zero constant. Designing good quantumcodes is highly nontrivial, just as it is in the classical case. The quantum Gilbert-Varshamovbound (QGVB) is a lower bound on an achievable relative distance of a quantum code ofa fixed rate. Explicit constructions of good quantum codes for q ≤ 7 have been studied[1, 2, 3], but they all fail to satisfy the QGVB. The QGVB bound is attainable for the familyof all random quantum codes [4], the family of random stabilizer codes [5], and the familyof random nondegenerate stabilizer codes [6], and the family of random degenerate stabilizercodes [6]. We show that concatenated quantum codes, with a quantum outer code havinga known structure and being efficiently decodable, and randomly chosen independent innerquantum codes also attains the QGVB. Thus our family of random quantum codes has morestructure than previously studied examples.In this paper, we generalize a special case of Thommesen’s result [7] to the quantumcase. He showed that a code from the family of binary concatenated codes made with a1
We show that for any R,ε>0R,ε>0R,ε>0 and any prime p≥2p\geq 2p≥2, there exists an infinite family of ppp-ary quantum low-density parity-check (QLDPC) codes, rate RRR, checks of weight Oε(1)O_ε(1)Oε​(1), and normalized distance at least δGV(p,R)−εδ_{\mathrm{GV}}(p,R)-εδGV​(p,R)−ε. Here, δGV(p,R)δ_{\mathrm{GV}}(p,R)δGV​(p,R) denotes the quantum Gilbert-Varshamov (GV) bound for ppp-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily \emph{degenerate}, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.