Aspects of mutually unbiased bases in odd-prime-power dimensions

We rephrase the Wootters-Fields construction [W. K. Wootters and B. C. Fields, Ann. Phys. 191, 363 (1989)] of a full set of mutually unbiased bases in a complex vector space of dimensions ${N=p}^r,$ where p is an odd prime, in terms of the character vectors of the cyclic group G of order p. This form may be useful in explicitly writing down mutually unbiased bases for ${N=p}^r.$