Coding theory: the unit-derived methodology

Hurley, Ted; Hurley, Donny

doi:10.1504/ijicot.2018.10013082

Cited by 8 publications

(76 citation statements)

References 14 publications

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“…These codes are defined using the unit-derived methods of [8]. The constructions in [9] are now used to define series of dual-containing codes, and Hermitian dualcontaining codes, from which mds quantum codes are constructed by the CSS constructions. The dualcontaining codes obtained have efficient decoding algorithms by [9] giving efficient decoding algorithms for the quantum codes constructed.…”

Section: The Dual-containing Codesmentioning

confidence: 99%

“…Theorem 2.1 [9] Let C be a code generated by taking any r rows of F n in arithmetic sequence with arithmetic difference k satisfying gcd(n, k) = 1. Then C is an mds (maximum distance separable) [n, r, n− r + 1] code.…”

Section: The Constructionsmentioning

confidence: 99%

“…Efficient decoding algorithms are in addition given in that paper [9]. The unit-derived strategy as in [12,8] is used for the constructions and analysis.…”

Section: The Constructionsmentioning

confidence: 99%

“…Then C ⊥ is generated by the rows excluding {e 0 , e 9 , e 8 , e 7 , e 6 , e 5 } and thus C ⊥ =< e 1 , e 2 , e 3 , e 4 >. Note in this example, by [9], that C is a [10,6,5] code and now we see that it contains its dual C ⊥ =< e 1 , e 2 , e 3 , e 4 >. By CSS construction a [ [10,2,5]] quantum mds code is obtained.…”

Section: The Constructionsmentioning

confidence: 99%

“…Codes from [9] which can be shown to be dual-containing relative to the Euclidean inner product are used to construct mds quantum codes by the CSS construction.…”

Section: The Constructionsmentioning

confidence: 99%

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Quantum error-correcting codes: the unit design strategy

Hurley

Hurley³

2018

IJICOT

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Series of maximum distance quantum error-correcting codes are developed and analysed. For a given rate and given error-correction capability, quantum error-correcting codes with these specifications are constructed. The codes are explicit with efficient decoding algorithms. For a given field maximum length quantum codes are constructed. IntroductionQuantum error-correcting codes have an important role in quantum computing and are used to protect quantum information from errors due to quantum noise and decoherence.A short introduction to quantum coding, with history and bibliographical notes, is given in Kim and Matthews [11]. Background information on quantum codes and quantum information theory may be found in the book by Nielsen and Chung [16]. A survey article with emphasis on topological aspects of quantum computing appears in Rowell and Wang [18].The literature on quantum error-correcting codes is massive and expanding. Required background on (classical) coding theory and on basic algebra, including in particular Field Theory, may be found in [3] or in [15]. GF (q) will denote the finite field of order q; of necessity q is a power of a prime. The non-zero elements of GF (q) form a cyclic group of order (q − 1) and any generator of the group is termed a primitive element of GF (q). A (classical) code of length n, dimension k and distance d over GF (q) is denoted by [n, k, d] q or simply by [n, k, d] when the field is understood or given.Seminal work of Calderbank, Shor and Steane [4,5,22] provide the relationship between classical codes and quantum error-correcting codes. Their construction is now known as the CSS construction. Following Rains' works [17] on nonbinary quantum codes, the work of Calderbank, Shor and Steane was extended to nonbinary cases by Ashikhmin and Knill, [2,7]. Gottesman [6] had previously developed the stabilizer formalism for quantum codes. See also work of Shor and Steane in [19,20,21].An [[n, k, d]] q code is a quantum code of length n, dimension k and minimum distance d over the field GF (q); the word q−ary code is sometimes used. When the field is understood or given, the notation [[n, k, d]] can be used, without the q suffix. Use [[n, k, ≥ d]] to mean a quantum code of length n, dimension k and minimum distance at least d.A classical [n, k, d] code satisfies the Singleton bound d ≤ (n − k + 1) and a code reaching this bound is called an mds (maximum distance separable) code. A quantum [[n, r, d]] code satisfies the quantum Singleton bound 2d ≤ (n − r + 2) and a quantum code attaining the bound is called an mds quantum code. The quantum codesHere series of mds quantum codes are constructed. For a given rate and given error-correcting capability quantum error-correcting mds codes of this rate and capability are given. Efficient decoding algorithms

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Section: The Dual-containing Codesmentioning

confidence: 99%

Section: The Constructionsmentioning

confidence: 99%

“…Efficient decoding algorithms are in addition given in that paper [9]. The unit-derived strategy as in [12,8] is used for the constructions and analysis.…”

Section: The Constructionsmentioning

confidence: 99%

Section: The Constructionsmentioning

confidence: 99%

“…Codes from [9] which can be shown to be dual-containing relative to the Euclidean inner product are used to construct mds quantum codes by the CSS construction.…”

Section: The Constructionsmentioning

confidence: 99%

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Quantum error-correcting codes: the unit design strategy

Hurley

Hurley³

2018

IJICOT

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Entanglement-assisted quantum error-correcting codes from units

Hurley¹,

Hurley²,

Hurley³

2018

Preprint

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Entanglement-assisted quantum error-correcting codes (EAQECCs) to desired rate, error-correcting capability and maximum shared entanglement are constructed. Thus for a required rate R, required error-correcting capability to correct t errors, mds (maximum distance separable) EAQECCs of the form [[n, r, d; c]] with R = r n , d ≥ (2t + 1), c = (n − r), d = (n − r + 1) are constructed. Series of such codes may be constructed where the rate and the relative distance approach non-zero constants as n approaches infinity. The codes may also be constructed over prime order fields in which modular arithmetic may be employed.

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Linear complementary dual, maximum distance separable codes

Hurley

2019

Preprint

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Linear complementary dual (LCD) maximum distance separable (MDS) codes are constructed to given specifications. For given n and r < n, with n or r (or both) odd, MDS LCD (n, r) codes are constructed over finite fields whose characteristic does not divide n. Series of LCD MDS codes are constructed to required rate and required error-correcting capability. Given the field GF (q) and n/(q − 1), LCD MDS codes of length n and dimension r are explicitly constructed over GF (q) for all r < n when n is odd and for all odd r < n when n is even. For given dimension and given errorcorrecting capability LCD MDS codes are constructed to these specifications with smallest possible length. Series of asymptotically good LCD MDS codes are explicitly constructed. Efficient encoding and decoding algorithms exist for all the constructed codes.Linear complementary dual codes have importance in data storage, communications' systems and security.

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Coding theory: the unit-derived methodology

Cited by 8 publications

References 14 publications

Quantum error-correcting codes: the unit design strategy

Quantum error-correcting codes: the unit design strategy

Entanglement-assisted quantum error-correcting codes from units

Linear complementary dual, maximum distance separable codes

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