Existence of APAV(q,k) with q a prime power ≡5 (mod 8) and k≡1 (mod 4)

Stinson introduced authentication perpendicular arrays APAλ(t,k,v), as a special kind of perpendicular arrays, to construct authentication and secrecy codes. Ge and Zhu introduced APAV(q,k) to study APA1(2,k,v) for k=5, 7. Chen and Zhu determined the existence of APAV(q,k) with q a prime power ≡3 (mod 4) and odd k>1. In this article, we show that for any prime power q≡5 (mod 8) and any k≡1 (mod 4) there exists an APAV(q,k) whenever q>((E+E2+4F)/2)2, where E=[(7k−23)m+3]25m−3, F=m(2m+1)(k−3)25m and m=(k−1)/4.