Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical $\mathsf{T}$ gates. As a non-Clifford operation, the $\mathsf{T}$ gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical $\mathsf{T}$ gates. As a non-Clifford operation, the $\mathsf{T}$ gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.

quant-pharXiv:2605.24519May 26, 20260 citations

Double quadratic residue codes and self-dual double cyclic codes

Arezoo Soufi Karbaski, Taher Abualrub, Steven T. Dougherty

Applicable Algebra in Engineering Communication and ComputingMay 29, 20201 citation

Cyclic triorthogonal codes in prime dimension

Shiroman Prakash

We study cyclic triorthogonal codes in prime dimension $p$ , on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd $p$ , we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to $p$ , triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for $p=3$ up to length $121$ and $p=5,7$ up to length $96$ . For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$ . For qudits, we find codes such as $[[19,1,3]]_3$ and $[[7,1,3]]_7$ , which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of $[[p,1,2\lfloor p/6\rfloor+1]]_p$ codes.We study cyclic triorthogonal codes in prime dimension $p$, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd $p$, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to $p$, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for $p=3$ up to length $121$ and $p=5,7$ up to length $96$. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length $763$. For qudits, we find codes such as $[[19,1,3]]_3$ and $[[7,1,3]]_7$, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of $[[p,1,2\lfloor p/6\rfloor+1]]_p$ codes.

quant-pharXiv:2610.08012Oct 7, 20260 citations

Lineare Codes

Anton Betten, Harald Fripertinger, Adalbert Kerber +2 more

Springer eBooksJan 1, 19980 citations

Code conversion with the quantum Golay code for a universal transversal gate set

Matthew Sullivan

The [[7,1,3]] Steane code and [[23,1,7]] quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates but require some other scheme for fault-tolerant $T$ gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal $T$ gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal $T$ gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.The [[7,1,3]] Steane code and [[23,1,7]] quantum Golay code have been identified as good candidates for fault-tolerant quantum computing via code concatenation. These two codes have transversal implementations of all Clifford gates but require some other scheme for fault-tolerant $T$ gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal $T$ gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal $T$ gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.

arXiv:2307.14425Physical Review AApr 18, 20247 citations

Triorthogonal codes are a basic ingredient in magic-state distillation, as they enable transversal implementation of logical

\mathsf{T}

gates. As a non-Clifford operation, the

\mathsf{T}

gate enables universal fault-tolerant quantum computation when combined with Clifford operations. We derive an existence criterion for even-weight triorthogonal generator matrices with a target dual minimum distance. The criterion combines triorthogonality constraints with MacWilliams identities via Krawtchouk-polynomial conditions on the dual weight distribution, yielding an integer linear programming formulation for the construction problem. We find novel nontrivial triorthogonal codes (degenerate and non-degenerate) that are not necessarily constructed from triply even binary linear codes. By imposing automorphism-group constraints, the formulation also recovers the known [[49,1,5]] and [[95,1,7]] triorthogonal codes. We also evaluate the performance of the obtained high-distance triorthogonal codes over the dephasing channel using different decoding methods, showing that automorphism ensemble decoding (AED) and guessing random additive-noise decoding (GRAND) are promising options.

We study cyclic triorthogonal codes in prime dimension

p

, on which a transversal third-level diagonal gate acts as a logical non-Clifford gate. For odd

p

, we use a definition of triorthogonality that does not require self-orthogonality. For cyclic codes of length coprime to

p

, triorthogonality reduces to a sumset condition on the spectral support, which makes exhaustive search feasible: for two natural constructions, we enumerate all maximal codes of this kind for

p=3

up to length

121

and

p=5,7

up to length

96

. For qubits, we prove that a cyclic code of odd length is triply even if and only if it is classically triorthogonal, giving a complete description of triply-even cyclic codes, and we determine all such codes up to length

763

. For qudits, we find codes such as

[[19,1,3]]_3

and

[[7,1,3]]_7

, which improve on the shortest Reed-Muller and Reed-Solomon codes in both overhead exponent and distillation threshold. The latter is the first member of an infinite family of

[[p,1,2\lfloor p/6\rfloor+1]]_p

codes.

T

gates. Using magic states, Clifford operations, and measurements is one common scheme, but magic-state distillation can have a large overhead. Code conversion is one avenue for implementing a universal gate set fault tolerantly without the use of magic-state distillation. Analogously to how the [[7,1,3]] Steane code can be fault tolerantly converted to and from the [[15,1,3]] Reed-Muller code which has a transversal

T

gate, the [[23,1,7]] Golay code can be converted to a [[95,1,7]] triorthogonal code with a transversal

T

gate. A crucial ingredient of this procedure is the [[49,1,5]] triorthogonal code, which can itself be seen as being related to the self-dual [[17,1,5]] two-dimensional color code. Additionally, a method for code conversion based on a transversal cnot between the codes, rather than stabilizer measurements, is described.