On the 2-ranks of a class of unitals

2015/10/12 by Rocco Trombetti, Yue Zhou, Trombetti, Rocco +1
Computer Science · Engineering · Mathematics · #11L05 #12K10 #51A35 #51A45 #Algebra over a field #Arithmetic #Binary number #Class (philosophy) #Coding theory and cryptography #Combinatorics #Combinatorics (math.CO) #Computer science #FOS: Mathematics #Finite Group Theory Research #Geometry #Kloosterman sum #Mathematical analysis #Mathematics #Order (exchange) #Physics #Plane (geometry) #Pure mathematics #Unital #Upper and lower bounds #graph theory and CDMA systems #math.CO #msc:11L05 #msc:12K10 #msc:51A35 #msc:51A45

paper · pdf · doi:10.48550/arxiv.1510.03340

arXiv admin note: text overlap with arXiv:1508.07279

arxiv created 2015/10/12 · openalex publication_date 2015/10/12 · arxiv updated 2015/10/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let U_θ be a unital defined in a shift plane of odd order q², which are constructed recently by the authors. In particular, when the shift plane is desarguesian, U_θ is a special Buekenhout-Metz unital formed by a union of ovals. We investigate the dimensions of the binary codes derived from U_θ. By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves the result obtained by Leung and Xiang in 2009. In particular, for q=3^m, this new lower bound equals (2)/(3)(q³+q²-2q)-1 for even m and (2)/(3)(q³+q²+q)-1 for odd m.

Citations

• Unitals in shift planes of odd order

Related

• Unitals in shift planes of odd order
• The feet of orthogonal Buekenhout-Metz unitals
• Unitals with many Baer secants through a fixed point
• New unitals in projective planes of order 16
• A classification of unitals in nearfield planes with maximal automorphism group