Codes over Z2m and Jacobi Forms of Genus n

Choie, YoungJu; Kim, Haesuk

doi:10.1006/jcta.2000.3168

Cited by 10 publications

(1 citation statement)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [16], codes over quaternion integers, where the elements are from the commutative ring of even integers, are given. Codes over quaternion integers, and codes over Lipschitz integers were studied in papers given in [13][14][15][16][17][18][19][20]. Without loss of generality in [15], [16], we consider Lipschitz integers, which their norm is both a prime integer and not a prime integer and each component is nonzero, and such that α 1 ≥ α 2 ≥ α 3 ≥ α 4 > 0 where gcd(α 1 , α 2 , α 3 , α 4 ) = 1 in this study.…”

Section: Introductionmentioning

confidence: 99%

Encoder Lipschitz Integers: The Lipschitz integers that have the ”division with small remainder” property

Duram

Güzeltepe

2021

Preprint

View full text Add to dashboard Cite

The residue class set of a Lipschitz integer is constructed by modulo function with primitive Lipschitz integer whose norm is a prime integer, i.e. prime Lipschitz integer. In this study, we consider primitive Lipschitz integer whose norm is both a prime integer and not a prime integer. If the norm of each element of the residue class set of a Lipschitz integer is less than the norm of the primitive Lipschitz integer used to construct the residue class set of the Lipschitz integer, then, the Euclid division algorithm works for this primitive Lipschitz integer. The Euclid division algorithm always works for prime Lipschitz integers. In other words, the prime Lipschitz integers have the ”division with small remainder” property. However, this property is ignored in some studies that have a constructed Lipschitz residue class set that lies on primitive Lipschitz integers whose norm is not a prime integer. In this study, we solve this problem by defining Lipschitz integers that have the ”division with small remainder” property, namely, encoder Lipschitz integers set. Therefore, we can define appropriate metrics for codes over Lipschitz integers. Also, we investigate the performances of Lipschitz signal constellations (the left residue class set) obtained by modulo function with Lipschitz integers, which have the ”division with small remainder” property, over the additive white Gaussian noise (AWGN) channel by agency of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

show abstract