2023/02/21 by Simone de Melo Costa, Costa, Simone, Lorenzo Mella +3 · 1 citation
Computer Science · Engineering · #05B20 #05B30 #05C10 #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2302.10617
openalex publication_date 2023/02/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2015, Archdeacon proposed the notion of Heffter arrays in view of its connection to several other combinatorial objects. In the same paper he also presented the following variant. A weak Heffter array WH(m,n;h,k) is an m × n matrix A such that: each row contains h filled cells and each column contains k filled cells; for every x ∈ ℤ2nk+1 ∖ \0\, there is exactly one cell of A whose element is one of the following: x,-x,± x,∓ x, where the upper sign on ± or ∓ is the row sign and the lower sign is the column sign; the elements in every row and column (with the corresponding sign) sum to 0 in ℤ2nk+1. Also the ``weak concept'', as the classical one, is related to several other topics, such as difference families, cycle systems and biembeddings. Many papers on Heffter arrays have been published, while no one on weak Heffter arrays has been written. This is the first one and here we explore necessary conditions, existence and non-existence results, and connections to biembeddings into non-orientable surfaces.