A hypercubic billiard word --or cubic billiard word in dimension d-- is a word on the d-letter alphabet denoting the sequence of the faces successively hit by a billiard ball moving in the unit cube of R^d , in which two parallel faces are encoded by the same letter. The imbalance of a word w is the maximal difference of the number of occurrences of a letter in two subwords of w of the same length. We know from the work of Morse and Hedlund (1940) that square billiard words generated by a momentum with rationally independent entries are exactly binary aperiodic words with imbalance equal to 1 (also known as Sturmian words). Vuillon (2003) showed that cubic billiard words whose momentum has rationally independent entries in dimension d have an imbalance lower than, or equal to, d − 1.