# On Characterization of Finite Geometric Distributive Lattices

@article{Basu2021OnCO, title={On Characterization of Finite Geometric Distributive Lattices}, author={Pranab Basu}, journal={ArXiv}, year={2021}, volume={abs/2109.07189} }

A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. Kötter and Kschischang proved that codes in the linear lattice can be used for error and erasure-correction in random networks. Codes in the linear lattice have previously been shown to be special cases of codes in modular lattices. Two well known classifications of modular lattices are geometric and distributive lattices. We have identified the…

## References

SHOWING 1-10 OF 33 REFERENCES

On the bounds of certain maximal linear codes in a projective space

- Mathematics, Computer Science2015 IEEE International Symposium on Information Theory (ISIT)
- 2015

The conjecture that the largest cardinality of a linear code, that contains F<sup>n</sup><sub>q</sub>, is 2 is proved and characterized and the maximal linear codes that contain F are characterized.

and A

- Vardy, “Linearity and complements in projective space,” Linear Algebra and its Applications, vol. 438, no. 1, pp. 57–70
- 2013

Coding for Errors and Erasures in Random Network Coding

- Mathematics, Computer ScienceIEEE Transactions on Information Theory
- 2008

A Reed-Solomon-like code construction, related to Gabidulin's construction of maximum rank-distance codes, is described and a Sudan-style ldquolist-1rdquo minimum-distance decoding algorithm is provided.

The Lattice Structure of Linear Subspace Codes

- Mathematics, Computer ScienceArXiv
- 2019

It is proved that a linear code in a projective space forms a sublattice of the corresponding projective lattice if and only if the code is closed under intersection.

On linear subspace codes closed under intersection

- Mathematics, Computer Science2015 Twenty First National Conference on Communications (NCC)
- 2015

It is shown that the conjecture that the largest cardinality of a linear subspace code in Pq(n) is 2n holds forlinear subspace codes that are closed under intersection, i.e., codes having the property that the intersection of any pair of codewords is also a codeword.

Equidistant Linear Codes in Projective Spaces

- Computer Science, MathematicsArXiv
- 2021

It is established that the problem of finding equidistant linear codes of maximum size in Pq(n) with constant distance 2d is equivalent to the problems of finding the largest d-intersecting family of subspaces in GQ(n, 2d) for all 1 ≤ d ≤ bn2 c.

Johnson Type Bounds for Mixed Dimension Subspace Codes

- Mathematics, Computer ScienceElectron. J. Comb.
- 2019

Improved upper bounds for subspace codes in random linear network coding are given based on the Johnson bound for constant dimension codes.

and S

- Kurz, “Johnson type bounds for mixed dimension subspace codes,” The Electronic Journal of Combinatorics, pp. P3–39
- 2019

Distributivity for Upper Continuous and Strongly Atomic Lattices

- Mathematics, Computer ScienceStud Logica
- 2017

It is proved that an upper continuous and strongly atomic lattice is distributive if and only if it satisfies (D) and ($$\hbox {D}^*$$D∗) which are strengthenings of Birkhoff's conditions.

A note on equidistant subspace codes

- Mathematics, Computer ScienceDiscret. Appl. Math.
- 2016

A classification of the largest 1-intersecting codes in PG ( 5 , 2 ) , whose codewords are planes, is provided and new constructions of large equidistant codes are presented.