Constructions of large cyclic constant dimension codes via Sidon spaces

Zhang, He; Tang, Chunming

doi:10.1007/s10623-022-01095-3

Cited by 10 publications

(4 citation statements)

References 19 publications

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“…Hoy, muchos investigadores publican resultados relacionados con algoritmos y métodos para construir espacios Sidon, además de los artículos mencionados, el lector puede consultar (Niu, Xiao, & Gao, 2022;Zhang & Cao, 2022;Zhang & Tang, 2023), y las referencias en ellos. El reciente concepto de Espacio Multi-Sidon (Zullo, 2021(Zullo, , 2023 puede considerarse como una generalización de la noción presentada en esta sección.…”

Section: Espacios Sidonunclassified

Conjuntos Sidon en Contextos Finitos

Trujillo Solarte

2023

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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Un conjunto A de enteros se llama conjunto Sidon si todas las sumas de dos elementos en A son distintas; es decir si para todo a, b, c, d ∈ A (a + b = c + d)⇒ {a,b} = {c,d}. Estos conjuntos los consideró el analista Simon Sidon, a principios de los años 1930, en sus investigaciones sobre análisis de Fourier. Una regla Golomb es un conjunto de enteros (no negativos) con la propiedad que todas las diferencias no cero son distintas. Estas reglas especiales aparecieron en el estudio de interferencias en radiofrecuencias, realizado por Wallace Babcock, a principios de los años 1950. En este artículo se realiza un recorrido a través de contextos finitos en los que intervienen estos objetos matemáticos. Los contextos considerados son: conjuntos finitos de números enteros, grupos cíclicos, retículos bidimensionales de coordenadas enteras (arreglos Costas, secuencias Sonar) y funciones APN. En buena parte, estas notas son resultado parcial de actividades realizadas durante mi actual año sabático en la Universidad del Cauca, y son también una versión ampliada de la conferencia invitada, que con el mismo título presenté durante el “55 Congreso Nacional de la Sociedad Matemática Mexicana”, realizado en la Universidad de Guadalajara del 23 al 28 de octubre de 2022 (https://www.smm.org.mx/congreso).

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Section: Espacios Sidonunclassified

Conjuntos Sidon en Contextos Finitos

Trujillo Solarte

2023

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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“…In the first family, we find the examples in [40,Constructions 11 and 15], [50,Lemma 3.1], [24, Lemmas 4.1 and 4.2] and [51,Lemma 2.4], in the second one we find examples in [53, Theorems 3.2 and 3.6], [50, Lemma 3.3, Theorems 3.5 and 3.13], [24,Theorems 4.5,4.7,4.9,, [49, Lemma 5 and Theorem 6], [52, Theorems 3.1 and 3.2] and [16,Theorem 3.3], and for the last family we can find examples in [9, Corollary 4 and Section IIIA] and [41,Section 7.3].…”

Section: Introductionmentioning

confidence: 99%

Constructions and equivalence of Sidon spaces

Castello,

Polverino,

Santonastaso

et al. 2023

J Algebr Comb

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Sidon spaces have been introduced by Bachoc et al. (in: Mathematical Proceedings of the Cambridge Philosophical Society, Cambridge University Press, 2017) as the q-analogue of Sidon sets. The interest on Sidon spaces has increased quickly, especially after the work of Roth et al. (IEEE Trans Inform Theory 64(6):4412–4422, 2017), in which they highlighted the correspondence between Sidon spaces and cyclic subspace codes. Up to now, the known constructions of Sidon Spaces may be divided in three families: the ones contained in the sum of two multiplicative cosets of a fixed subfield of $$\mathbb {F}_{q^n}$$ F q n , the ones contained in the sum of more than two multiplicative cosets of a fixed subfield of $$\mathbb {F}_{q^n}$$ F q n and finally the ones obtained as the kernel of subspace polynomials. In this paper, we will mainly focus on the first class of examples, for which we provide characterization results and we will show some new examples, arising also from some well-known combinatorial objects. Moreover, we will give a quite natural definition of equivalence among Sidon spaces, which relies on the notion of equivalence of cyclic subspace codes and we will discuss about the equivalence of the known examples.

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“…In this paper, we will survey new constructions of cyclic constant dimension codes including multiple orbits from high-dimensional Sidon spaces ( see Theorems 3.7 and 3.10 ). More results and applications about constant dimension codes via Sidon spaces can be found in [6,7,16,17].…”

Section: Introductionmentioning

confidence: 99%

Three new families of large cyclic subspace codes via high-dimensional Sidon spaces

Zhang

Tang

2022

Preprint

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k-dimensional subspace codes, also called as constant dimension codes, play a crucial role in error correction of random network coding for ten years. Especially, large cyclic constant dimension codes with the optimal minimum distance are possible candidates owing to efficient encoding and decoding algorithms. The seminal work of Ben-Sasson et al. was using the roots of subspace polynomials to construct cyclic subspace codes. Shortly after, Roth et al. introduced the definitions of max-span and min-span Sidon spaces and designed a kind of new cyclic subspace codes which contain multiple orbits from Sidon spaces. By exploring Roth's ideas, Niu et al. and Li et al. further investigated high-dimensional Sidon spaces employing the direct sum to extend the previous results. However, we find that they only constructed high-dimensional Sidon spaces and presented several examples of cyclic subspace codes from Sidon spaces with dimension k. Combining the orbit of Sidon spaces with dimension k or more than k respectively, we will provide three families of larger cyclic subspace codes. Additionally, our constructions of cyclic subspace codes can yet maintain their the optimality of minimum distance.

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Constructions of large cyclic constant dimension codes via Sidon spaces

Cited by 10 publications

References 19 publications

Conjuntos Sidon en Contextos Finitos

Conjuntos Sidon en Contextos Finitos

Constructions and equivalence of Sidon spaces

Three new families of large cyclic subspace codes via high-dimensional Sidon spaces

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