Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On Mixed Codes with Covering Radius 1 and Minimum Distance 2. | On Mixed Codes with Covering Radius 1 and Minimum Distance 2 Wolfgang Haas Albert-Ludwigs-Universitat Mathematisches Institut Eckerstr. 1 79104 Freiburg Germany wolfgang_haas@ Jorn Quistorff Department 4 FHTW Berlin University of Applied Sciences 10313 Berlin Germany Submitted Mar 13 2007 Accepted Jul 4 2007 Published Jul 19 2007 Mathematics Subject Classifications 94B60 94B65 05B15 Abstract Let R S and T be finite sets with R r S s and TI t. A code C c R X S X T with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality K r s t 2 . These bounds turn out to be best possible in many instances. Focussing on the special case t s we determine K r s s 2 when r divides s when r s 1 when s is large relative to r when r is large relative to s as well as K 3r 2r 2r 2 . Some open problems are posed. Finally a table with bounds on K r s s 2 is given. 1 Introduction Let Q denote a finite alphabet with IQI q 2. The Hamming distance d y y between y y0 2 Qn denotes the number of coordinates in which y and y0 differ. For y 2 Qn and C c Qn with C 0 we set d y C minx2C d y x . We say that y is R-covered by C if d y C R and that C c Qn is R-covered by C if every y 2 C is R-covered by C. A code C c Qn of length n has covering radius at most R if Qn is R-covered by C. C has minimum distance at least d when any two distinct codewords have Hamming distance at least d. Combinatorial coding theory deals with Aq n d the maximal cardinality of a THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 R51 1 code C c Qn with minimum distance d and Kq n R the minimal cardinality of a code C c Qn with covering radius R see 2 . q-ary codes with covering radius at most 1 and minimum distance at least 2 as well as the corresponding non-extendable partial multiquasigroups have been studied in 9 7 8 1 6 . Equivalent .
