Recognition and Analysis of Image Patterns Based on Latin Squares by Means of Computational Algebraic Geometry

With the particular interest of being implemented in cryptography, the recognition and analysis of image patterns based on Latin squares has recently arisen as an efficient new approach for classifying partial Latin squares into isomorphism classes. This paper shows how the use of a Computer Algebra System (CAS) becomes necessary to delve into this aspect. Thus, the recognition and analysis of image patterns based on these combinatorial structures benefits from the use of computational algebraic geometry to determine whether two given partial Latin squares describe the same affine algebraic set. This paper delves into this topic by focusing on the use of a CAS to characterize when two partial Latin squares are either partial transpose or partial isotopic.

Restricted completion of sparse partial Latin squares

Ann×npartial Latin squarePis calledα-dense if each row and column has at mostαnnon-empty cells and each symbol occurs at mostαntimes inP. Ann×narrayAwhere each cell contains a subset of {1,…,n} is a (βn,βn, βn)-array if each symbol occurs at mostβntimes in each row and column and each cell contains a set of size at mostβn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constantsα,β> 0 such that, for every positive integern, ifPis anα-densen×npartial Latin square,Ais ann×n (βn, βn, βn)-array, and no cell ofPcontains a symbol that appears in the corresponding cell ofA, then there is a completion ofPthat avoidsA; that is, there is a Latin squareLthat agrees withPon every non-empty cell ofP, and, for eachi,jsatisfying 1 ≤i,j≤n, the symbol in position (i,j) inLdoes not appear in the corresponding cell ofA.

Completing Partial Latin Squares with One Nonempty Row, Column, and Symbol

Let $r,c,s\in\{1,2,\ldots,n\}$ and let $P$ be a partial latin square of order $n$ in which each nonempty cell lies in row $r$, column $c$, or contains symbol $s$. We show that if $n\notin\{3,4,5\}$ and row $r$, column $c$, and symbol $s$ can be completed in $P$, then a completion of $P$ exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row $r$, column $c$, and symbol $s$ can be completed.