Short covering codes in Hamming spaces Anderson N. Martinh ao - - PowerPoint PPT Presentation

Short covering codes in Hamming spaces

Anderson N. Martinh˜ ao

Centro de Ciˆ encias Exatas Universidade Estadual de Maring´ a Programa de P´

• s-Graduac

¸˜ ao em Matem´ atica Joint work with Dr. Emerson L. Monte Carmelo

January 28, 2015

Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 1 / 6

Definition Given two vectors u = (u1, . . . , un), and v = (v1, . . . , vn) in Zn

q , the Hamming

distance between u and v is the number d(u, v) = |{i : ui = vi}|. Definition Let u ∈ Zn

q . The ball of center u and radius R is the subset

B(u, R) = {v ∈ Zn

q : d(u, v) ≤ R}.

Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 2 / 6

Classical coverings

Definition A subset C of Zn

q is a covering code of Zn q if

Zn

q =

• c∈C

B(c, R). A natural question is: What is the minimum cardinality Kq(n, R) of a covering code of Zn

q ?

An example that demonstrates how difficult it is to obtain exact classes is the fact that the value Kq(4, 1) has not yet been determined. Indeed, 115 ≤ K7(4, 1) ≤ 123.

Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 3 / 6

Short Coverings

Consider Fq a finite field with q elements. Definition Let u ∈ F3

• q. The extended ball of center u and radius 1 is

E(u) =

• λ∈Fq

B(λu, 1). Definition A subset H of F3

q is a short covering when:

F3

q =

• h∈H

E(h) =

• λ∈Fq
• h∈H

B(λh, 1).

Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 4 / 6

What is the minimum cardinality c(q) of a short covering code of F3

q?

Due to the difficulty this problem are known only for some values c(q). q 2 3 4 5 7 8 9 c(q) 1 3 3 4 4–5 5–9 5–7

Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 5 / 6

References

• G. Cohen, I. Honkala, S. Litsyn, and A. Lobstein, Covering Codes,

North-Holland, Amsterdam, (1997). A.N. Martinh˜ ao, and E.L. Monte Carmelo, Short covering codes arising from matchings in weighted graphs. Mathematics of Computation, AMS, 82 (2013), 605–616. Taussky, Todd, J., Covering Theorems for Groups. Ann. Soc. Polonaise Math. 21, (1948), 303 - 305.

Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 6 / 6