November 2017, 11(4): 757-765. doi: 10.3934/amc.2017055

On primitive constant dimension codes and a geometrical sunflower bound

1. 

Departament d'Enginyeria de la Informacio i de les Comunicacions, Edifici Q, Universitat Autónoma de Barcelona, 08193 -Bellaterra -Cerdanyola del Vallés (Barcelona), Spain

2. 

Department of Mathematics (WE01), Krijgslaan 281 -building S25, 9000 Gent, Belgium

Received  April 2016 Published  November 2017

Fund Project: The fourth author is supported by a postdoctoral grant of the FWO-Flanders

In this paper we study subspace codes with constant intersection dimension (SCIDs). We investigate the largest possible dimension spanned by such a code that can yield non-sunflower codes, and classify the examples attaining equality in that bound as one of two infinite families. We also construct a new infinite family of primitive SCIDs.

Citation: Roland D. Barrolleta, Emilio Suárez-Canedo, Leo Storme, Peter Vandendriessche. On primitive constant dimension codes and a geometrical sunflower bound. Advances in Mathematics of Communications, 2017, 11 (4) : 757-765. doi: 10.3934/amc.2017055
References:
[1]

D. Bartoli and F. Pavese, A note on equidistant subspace codes, Discrete Appl. Math., 198 (2016), 291-296. doi: 10.1016/j.dam.2015.06.017.

[2]

A. BeutelspacherJ. Eisfeld and J. Müller, On sets of planes in ${\text{PG}}(d,q)$ intersecting mutually in one point, Geom. Dedicata, 78 (1999), 143-159. doi: 10.1023/A:1005294416997.

[3]

J. Eisfeld, On sets of $n$-dimensional subspaces of projective spaces intersecting mutually in an $(n-2)$-dimensional subspace, Discrete Math., 255 (2002), 81-85. doi: 10.1016/S0012-365X(01)00390-9.

[4]

T. Etzion and N. Raviv, Equidistant codes in the Grassmannian, Discrete Appl. Math., 186 (2015), 87-97. doi: 10.1016/j.dam.2015.01.024.

[5]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

[6]

K. Metsch and L. Storme, Partial $t$-spreads in ${\text{PG}}(2t+1,q)$, Des. Codes Cryptogr., 18 (1999), 199-216. doi: 10.1023/A:1008305824113.

[7]

B. Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76. doi: 10.1007/BF02410047.

show all references

References:
[1]

D. Bartoli and F. Pavese, A note on equidistant subspace codes, Discrete Appl. Math., 198 (2016), 291-296. doi: 10.1016/j.dam.2015.06.017.

[2]

A. BeutelspacherJ. Eisfeld and J. Müller, On sets of planes in ${\text{PG}}(d,q)$ intersecting mutually in one point, Geom. Dedicata, 78 (1999), 143-159. doi: 10.1023/A:1005294416997.

[3]

J. Eisfeld, On sets of $n$-dimensional subspaces of projective spaces intersecting mutually in an $(n-2)$-dimensional subspace, Discrete Math., 255 (2002), 81-85. doi: 10.1016/S0012-365X(01)00390-9.

[4]

T. Etzion and N. Raviv, Equidistant codes in the Grassmannian, Discrete Appl. Math., 186 (2015), 87-97. doi: 10.1016/j.dam.2015.01.024.

[5]

R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.

[6]

K. Metsch and L. Storme, Partial $t$-spreads in ${\text{PG}}(2t+1,q)$, Des. Codes Cryptogr., 18 (1999), 199-216. doi: 10.1023/A:1008305824113.

[7]

B. Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76. doi: 10.1007/BF02410047.

Figure 1.  The $(k,k-t)$-SCID described in Example 1
Figure 2.  The $(k,k-t)$-SCID described in Example 2
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