A Complete Classification of Doubly Even Self-dual Codes of Length 40

Betsumiya, Koichi; Harada, Masaaki; Munemasa, Akihiro

doi:10.37236/2593

Cited by 32 publications

(83 citation statements)

References 23 publications

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“…There are 16, 470 inequivalent extremal doubly even self-dual codes of length 40 [1]. We verified that 15, 940 of them contain no triple T such that ij has a clique of size 8 for all i, j ∈ {1, 2, .…”

Section: The Search Methodsmentioning

confidence: 83%

“…Two codes are equivalent if one can be obtained from the other by permuting the coordinates. A classification of doubly even self-dual codes was done for lengths up to 40 (see [1] and the references given therein, and the data can be found in [7]). For example, there are 16, 470 inequivalent extremal doubly even self-dual codes of length 40.…”

Section: Quasi-symmetric Designs and Doubly Even Self-dual Codesmentioning

confidence: 99%

“…Suppose that v ≡ 5 (mod 8), k ≡ 1 (mod 4), and x, y are odd. Let A be the incidence matrix of D. Let C be a doubly even self-dual code of length v + 3 containing the row span of the matrix (1). Then C has minimum weight at least (b + r)/r.…”

Section: Remarkmentioning

confidence: 99%

“…Let A be the incidence matrix of a quasi-symmetric 2-(37, 9, 8) design D with intersection numbers 1 and 3. Let C be a doubly even self-dual [40,20] code containing the row span of the matrix (1). Then C has minimum weight 8, and there is no codeword of weight 8 whose support contains the last three coordinates.…”

Section: Quasi-symmetric 2-(37 9 8) Designs and Doubly Even Self-dumentioning

confidence: 99%

“…The main result of this paper is the following theorem, which was established by using the recent classification of binary doubly even self-dual codes of length 40 [1]. Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

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Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

Harada

Munemasa

Tonchev

2017

J of Combinatorial Designs

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We prove that a certain binary linear code associated with the incidence matrix of a quasi-symmetric 2-(37, 9, 8) design with intersection numbers 1 and 3 must be contained in an extremal doubly even self-dual code of length 40. Using the classification of extremal doubly even self-dual codes of length 40, we show that a quasi-symmetric 2-(37, 9, 8) design with intersection numbers 1 and 3 does not exist.

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Section: The Search Methodsmentioning

confidence: 83%

Section: Quasi-symmetric Designs and Doubly Even Self-dual Codesmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Quasi-symmetric 2-(37 9 8) Designs and Doubly Even Self-dumentioning

confidence: 99%

“…The main result of this paper is the following theorem, which was established by using the recent classification of binary doubly even self-dual codes of length 40 [1]. Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

Harada

Munemasa

Tonchev

2017

J of Combinatorial Designs

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Narain CFTs and quantum codes at higher genus

Henriksson

Kakkar

McPeak

2023

J. High Energ. Phys.

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Code CFTs are 2d conformal field theories defined by error-correcting codes. Recently, Dymarsky and Shapere generalized the construction of code CFTs to include quantum error-correcting codes. In this paper, we explore this connection at higher genus. We prove that the higher-genus partition functions take the form of polynomials of higher-weight theta functions, and that the higher-genus modular group acts as simple linear transformations on these polynomials. We explain how to solve the modular constraints explicitly, which we do for genus 2. The result is that modular invariance at genus 1 and genus 2 is much more constraining than genus 1 alone. This allows us to drastically reduce the space of possible code CFTs. We also consider a number of examples of “isospectral theories” — CFTs with the same genus 1 partition function — and we find that they have different genus 2 partition functions. Finally, we make connection to some 2d CFTs known from the modular bootstrap. The n = 4 theory conjectured to have the largest possible gap in Virasoro characters, the SO(8) WZW model, is a code CFT, allowing us to give an expression for its genus 2 partition function. We also find some other known CFTs which are not code theories but whose partition functions satisfy the same simple polynomial ansatz as the code theories. This leads us to speculate about the usefulness of the code polynomial form beyond the study of code CFTs.

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Classical codes and chiral CFTs at higher genus

Henriksson

Kakkar

McPeak

2022

J. High Energ. Phys.

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Higher genus modular invariance of two-dimensional conformal field theories (CFTs) is a largely unexplored area. In this paper, we derive explicit expressions for the higher genus partition functions of a specific class of CFTs: code CFTs, which are constructed using classical error-correcting codes. In this setting, the Sp(2g, ℤ) modular transformations of genus g Riemann surfaces can be recast as a simple set of linear maps acting on 2g polynomial variables, which comprise an object called the code enumerator polynomial. The CFT partition function is directly related to the enumerator polynomial, meaning that solutions of the linear constraints from modular invariance immediately give a set of seemingly consistent partition functions at a given genus. We then find that higher genus constraints, plus consistency under degeneration limits of the Riemann surface, greatly reduces the number of possible code CFTs. This work provides a step towards a full understanding of the constraints from higher genus modular invariance on 2d CFTs.

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A Complete Classification of Doubly Even Self-dual Codes of Length 40

Abstract: A complete classification of binary doubly even self-dual codes of length 40 is given. As a consequence, a classification of binary extremal self-dual codes of length 38 is also given. * This work was supported by JST PRESTO program.

Cited by 32 publications

References 23 publications

Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

Self‐Dual Codes and the Nonexistence of a Quasi‐Symmetric 2‐(37,9,8) Design with Intersection Numbers 1 and 3

Narain CFTs and quantum codes at higher genus

Classical codes and chiral CFTs at higher genus

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