Abstract
Recently, the construction of Narain CFT from a certain class of quantum error correcting codes has been discovered. In particular, the spectral gap of Narain CFT corresponds to the binary distance of the code, not the genuine Hamming distance. In this paper, we show that the binary distance is identical to the so-called EPC distance of the boolean function uniquely associated with the quantum code. Therefore, seeking Narain CFT with large spectral gap can be addressed by getting a boolean function with high EPC distance. Furthermore, this problem can be undertaken by finding lower Peak-to-Average Power ratio (PAR) with respect to the binary truth table of the boolean function. Though this is neither sufficient nor necessary condition for high EPC distance, we construct some examples of relatively high EPC distances referring to the constructions for lower PAR. We also see that codes with high distance are related to induced graphs with low independence numbers.
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References
L. Dolan, P. Goddard and P. Montague, Conformal field theories, representations and lattice constructions, Commun. Math. Phys. 179 (1996) 61 [hep-th/9410029] [INSPIRE].
A. Dymarsky and A. Shapere, Quantum stabilizer codes and lattices and CFTs, JHEP 03 (2021) 160 [arXiv:2009.01244] [INSPIRE].
F.J. MacWilliams, N.J.A. Sloane and J.G. Thompson, Good self dual codes exist, Discrete Math. 3 (1972) 153
D.E. Muller, Application of Boolean algebra to switching circuit design and to error detection, Trans. I.R.E. Prof. Group Electron. Comput. EC-3 (1954) 6.
T. Hartman, C.A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
A. Dymarsky and A. Shapere, Comments on the holographic description of Narain theories, JHEP 10 (2021) 197 [arXiv:2012.15830] [INSPIRE].
L.E. Danielsen, On Self-Dual Quantum Codes, Graphs, and Boolean Functions, quant-ph/0503236 [INSPIRE].
K.S. Narain, New Heterotic String Theories in Uncompactified Dimensions < 10, Phys. Lett. B 169 (1986) 41 [INSPIRE].
S. Yahagi, Narain CFTs and error-correcting codes on finite fields, JHEP 08 (2022) 058 [arXiv:2203.10848] [INSPIRE].
L.E. Danielsen, T.A. Gulliver and M.G. Parker, Aperiodic propagation criteria for Boolean functions, Inf. Comput. 204 (2006) 741.
M. Hein, W. Dür, J. Eisert, R. Raussendorf, M. Van den Nest and H.-J. Briegel, Entanglement in Graph States and its Applications, quant-ph/0602096.
C. Riera and M.G. Parker, Generalized Bent Criteria for Boolean Functions (I), IEEE Trans. Inf. Theory 52 (2006) 4142.
L.E. Danielsen and M.G. Parker, Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with respect to the I, H, Nn Transform, in Sequences and Their Applications — SETA 2004, Seoul, South Korea (2004) [Lect. Notes Comput. Sci. 3486 (2005) 373] [cs/0504102].
B. Preneel, W. Van Leekwijck, L. Van Linden, R. Govaerts and J. Vandewalle, Propagation Characteristics of Boolean Functions, in Advances in Cryptology — EUROCRYPT’90, Aarhus, Denmark (1990) [Lect. Notes Comput. Sci. 473 (1991) 161].
C. Carlet, On cryptographic propagation criteria for boolean functions, Inf. Comput. 151 (1999) 32
B. Preneel, Analysis and Design of Cryptographic Hash Functions, Ph.D. Thesis, Katholieke Universiteit Leuven, Leuven Belgium (2003).
M.G. Parker and C. Tellambura, A construction for binary sequence sets with low peak-to-average power ratio, in Proceedings IEEE International Symposium on Information Theory, Lausanne, Switzerland (2002), pg. 239.
C. Riera and M.G. Parker, One and Two-Variable Interlace Polynomials: A Spectral Interpretation, in International Workshop on Coding and Cryptography — WCC 2005, Bergen, Norway (2005) [Lect. Notes Comput. Sci. 3969 (2006) 397].
M. Grassl and M. Harada, New self-dual additive 𝔽4-codes constructed from circulant graphs, Discrete Math. 340 (2017) 399 [arXiv:1509.04846] [INSPIRE].
M. Harada, New quantum codes constructed from some self-dual additive 𝔽4-codes, Inf. Process. Lett. 138 (2018) 35 [arXiv:1805.12229].
M. Buican, A. Dymarsky and R. Radhakrishnan, Quantum Codes, CFTs, and Defects, arXiv:2112.12162 [INSPIRE].
A. Dymarsky and A. Shapere, Solutions of modular bootstrap constraints from quantum codes, Phys. Rev. Lett. 126 (2021) 161602 [arXiv:2009.01236] [INSPIRE].
A. Dymarsky and A. Sharon, Non-rational Narain CFTs from codes over F4, JHEP 11 (2021) 016 [arXiv:2107.02816] [INSPIRE].
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Furuta, Y. Relation between spectra of Narain CFTs and properties of associated boolean functions. J. High Energ. Phys. 2022, 146 (2022). https://doi.org/10.1007/JHEP09(2022)146
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DOI: https://doi.org/10.1007/JHEP09(2022)146