On the gaps of the spectrum of volumes of trades. Issue 3 (1st November 2017)

• Record Type: Journal Article
• Authors: Krotov, Denis S.
• Is Part Of: Journal of combinatorial designs. Volume 26:Issue 3(2018:Mar.)
• Page Start: 119
• Page End: 126
• Publication Date: 2017-11-01
• Subjects: minimum volume -- Reed–Muller code -- t‐design -- trade; Combinatorial designs and configurations -- Periodicals; Configurations et schémas combinatoires -- Périodiques; 511.6
• DOI: 10.1002/jcd.21592
• Languages: English
• ISSNs: 1063-8539

Abstract

Abstract: A pair { T 0, T 1 } of disjoint collections of k ‐subsets (blocks) of a set V of cardinality v is called a t ‐ ( v, k ) trade or simply a t ‐trade if every t ‐subset of V is included in the same number of blocks of T 0 and T 1 . The cardinality of T 0 is called the volume of the trade. Using the weight distribution of the Reed–Muller code, we prove the conjecture that for every i from 2 to t, there are no t ‐trades of volume greater than 2 t + 1 − 2 i and less than 2 t + 1 − 2 i − 1 and derive restrictions on the t ‐trade volumes that are less than 2 t + 1 + 2 t − 1 .