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´ os-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 = ( u 1 , . . . , u n ) , and v = ( v 1 , . . . , v n ) in Z n q , the Hamming distance between u and v is the number d ( u, v ) = |{ i : u i � = v i }| . Definition Let u ∈ Z n q . The ball of center u and radius R is the subset B ( u, R ) = { v ∈ Z n q : d ( u, v ) ≤ R } . Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 2 / 6

Classical coverings Definition A subset C of Z n q is a covering code of Z n q if � Z n q = B ( c, R ) . c ∈C A natural question is: What is the minimum cardinality K q ( n, R ) of a covering code of Z n q ? An example that demonstrates how difficult it is to obtain exact classes is the fact that the value K q (4 , 1) has not yet been determined. Indeed, 115 ≤ K 7 (4 , 1) ≤ 123 . Anderson N. Martinh˜ ao () SP Coding School January 28, 2015 3 / 6

Short Coverings Consider F q a finite field with q elements. Definition Let u ∈ F 3 q . The extended ball of center u and radius 1 is � E ( u ) = B ( λu, 1) . λ ∈ F q Definition A subset H of F 3 q is a short covering when: F 3 � � � q = E ( h ) = B ( λh, 1) . h ∈H λ ∈ F q h ∈H 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 F 3 q ? Due to the difficulty this problem are known only for some values c ( q ) . 2 3 4 5 7 8 9 q 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