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Improved Quantum Codes from Metacirculant Graphs via Self-Dual Additive F 4 -Codes.

Padmapani Seneviratne, Martianus Frederic Ezerman

Published May 26, 2021• 1 citation
OpenAlex

Abstract

We use symplectic self-dual additive codes over $\mathbb{F}_4$ obtained from metacirculant graphs to construct, for the first time, $[[\ell, 0, d ]]$ qubit codes with parameters $(\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,21)\}$. Secondary constructions applied to the qubit codes result in many qubit codes that perform better than the previous best-known.

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Improved Quantum Codes from Metacirculant Graphs via Self-Dual Additive $\mathbb{F}_4$-Codes

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Circulant Based Extremal Additive Self-Dual Codes Over>tex<$hboxGF,(4)$>/tex<

Thomas Aaron Gulliver, J.-L. Kim

It is well known that the problem of finding stabilizer quantum-error-correcting codes (QECC) is transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to classify the extremal additive circulant self-dual codes of lengths up to 15, and construct good codes for lengths 16/spl les/n/spl les/27. We also classify the extremal additive 4-circulant self-dual codes of lengths 4,6,8,12,14, and 16 and most codes of length 10, and construct good codes of even lengths up to 22. Furthermore, we classify the extremal additive bordered 4-circulant self-dual codes of lengths 3,5,7,9,11,13,15, and 17, and construct good codes for lengths 19,21,23, and 25. We give the current status of known extremal (or optimal) additive self-dual codes of lengths 12 to 27.

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We use symplectic self-dual additive codes over F4\mathbb{F}_4F4​ obtained from metacirculant graphs to construct, for the first time, [[ℓ,0,d]][[\ell, 0, d ]][[ℓ,0,d]] qubit codes with parameters (ℓ,d)∈{(78,20),(90,21),(91,22),(93,21),(96,21)}(\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,21)\}(ℓ,d)∈{(78,20),(90,21),(91,22),(93,21),(96,21)}. Secondary constructions applied to the qubit codes result in many qubit codes that perform better than the previous best-known.
Improved Quantum Codes from Metacirculant Graphs via Self-Dual Additive F4\mathbb{F}_4F4​-Codes
We use symplectic self-dual additive codes over F4\mathbb{F}_4F4​ obtained from metacirculant graphs to construct, for the first time, [[ℓ,0,d]][[\ell, 0, d ]][[ℓ,0,d]] qubit codes with parameters (ℓ,d)∈{(78,20),(90,21),(91,22),(93,21),(96,21)}(\ell,d) \in \{(78, 20), (90, 21), (91, 22), (93,21),(96,21)\}(ℓ,d)∈{(78,20),(90,21),(91,22),(93,21),(96,21)}. Secondary constructions applied to the qubit codes result in many qubit codes that perform better than the previous best-known.
New quantum codes from metacirculant graphs via self-dual additive F4\mathbb{F}_4F4​-codes
We use multidimensional circulant approach to construct new qutrit stabilizer $\dsb{\ell, 0, d}$ codes with parameters (ℓ,d)∈{(51,16),(52,16),(54,17),(55,17),(57,17)}(\ell, d) \in \{(51, 16), (52, 16), (54, 17), (55, 17), (57, 17)\}(ℓ,d)∈{(51,16),(52,16),(54,17),(55,17),(57,17)} through symplectic self-dual additive codes over $\F_9$. In addition to these five new codes, we use bordered construction to derive two more qutrit codes with parameters (ℓ,d)∈{(53,16),(56,17)}(\ell, d) \in \{(53, 16), (56, 17)\}(ℓ,d)∈{(53,16),(56,17)} that improve upon previously best known parameters.
It is established in literature that finding stabilizer quantum error correcting codes (QECCS) is the same as finding self-dual additive codes over the finite field F4\mathbb{F}_{4}F4​ under the Hermitian trace inner product. Additionally, every self-dual additive code can be represented by a graph adjacency matrix. Many selfdual additive codes are constructed from circulant graphs. We introduce new graph code constructions: the Toeplitz Graph, the Multidimensional Toeplitz Graph (MDT), and the Generalized Toeplitz (GT) graph constructions. We consider some of the properties of the Toeplitz and Multidimensional Toeplitz Graphs and compare the Circulant and GT code constructions.
Abstract We introduce quaternary modified four μ\mu μ μ -circulant codes as a modification of four circulant codes. We give basic properties of quaternary modified four μ\mu μ μ -circulant Hermitian self-dual codes. We also construct quaternary modified four μ\mu μ μ -circulant Hermitian self-dual codes having large minimum weights. Two quaternary Hermitian self-dual [56, 28, 16] codes are constructed for the first time. These codes improve the previously known lower bound on the largest minimum weight among all quaternary (linear) [56, 28] codes. In addition, these codes imply the existence of a quantum [[56, 0, 16]] code.
Circulant Based Extremal Additive Self-Dual Codes Over>tex<hboxGF,(4)hboxGF,(4)hboxGF,(4)>/tex<
It is well known that the problem of finding stabilizer quantum-error-correcting codes (QECC) is transformed into the problem of finding additive self-orthogonal codes over the Galois field GF(4) under a trace inner product. Our purpose is to classify the extremal additive circulant self-dual codes of lengths up to 15, and construct good codes for lengths 16/spl les/n/spl les/27. We also classify the extremal additive 4-circulant self-dual codes of lengths 4,6,8,12,14, and 16 and most codes of length 10, and construct good codes of even lengths up to 22. Furthermore, we classify the extremal additive bordered 4-circulant self-dual codes of lengths 3,5,7,9,11,13,15, and 17, and construct good codes for lengths 19,21,23, and 25. We give the current status of known extremal (or optimal) additive self-dual codes of lengths 12 to 27.