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

Stinson introduced authentication perpendicular arrays APA _λ(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 + √E ² + 4F)/2)², where E = [(7k - 23)m + 3]2^5m - 3, F = m(2m + 1)(k - 3)2^5m and m = (k - 1)/4.