Binary code imbalance and two measures of average sidelobe level

The relationship between binary code imbalance and each of two measures of average autocorrelation sidelobe level is examined. For the first one, the arithmetic mean of signed sidelobes, an especially simple relationship is derived. It is found that the size of this average is minimized when imbalance is as dose as possible to the square root of the code length. The second is the square root of the average mean square sidelobe, where the average is taken over codes having the same imbalance. For this second measure, it is found that for any code length N ⩾ 4, there are three extreme points: a local maximum at zero imbalance and two global minima at ±U₀ for some U₀ > 0 that depends on N. Formulae are derived for U₀ in terms of N valid for any integer N ⩾ 4. These formulae behave like (3/2)√N asymptotically as N increases