Here we consider the set S(n;t,s) of length n words a₁a₂···a_n over the alphabet consisting of the elements of the field F₄ that have trace t and subtrace s. The trace of a word is the sum of its digits over the field F₄ , i.e. t = a₁+a₂+ ··· +a_n. The subtrace is the sum of the products of all n(n-1)/2 pairs of digits taken over the field F₄ , i.e. s = SUM( a_ia_j : 1 < i < j < n ).

Below we use x = RootOf( z²+z+1 ) and y = 1+x.

Examples:

• The one 4-ary string of trace 0, subtrace 1 and length 2 is { 11 }.
• The two 4-ary strings of trace x, subtrace y and length 2 are { 1y, y1 }.
• The three 4-ary strings of trace y, subtrace 1 and length 3 are { 11y, 1y1, y11 }.

Further Notes:

• The number S(n;t,s) can be computed from the following recurrence relation

  S(n;t,s) = S(n-1;t,s) + S(n-1;t-1,s-(t-1)) + S(n-1;t-x,s-x(t-x)) + S(n-1;t-y,s-y(t-y))

              = S(n-1;t,s) + S(n-1;t+1,s+t+1) + S(n-1;t+x,s+x(t+x)) + S(n-1;t+y,s+y(t+y))

  Note that all operations involving operands t or s are carried out over GF(4).

• Column (0,0) is sequence A073995 in Neil J. Sloane's database of integer sequences.
• Column (0,1),(0,x),(0,y) is sequence A073996 in Neil J. Sloane's database of integer sequences.
• Column (1,0),(x,0),(y,0) is sequence A073997 in Neil J. Sloane's database of integer sequences.
• Column (1,1),(x,y),(y,x) is sequence A073998 in Neil J. Sloane's database of integer sequences.
• Column (1,x),(x,1),(y,1),(1,x),(x,x),(y,y) is sequence A073999 in Neil J. Sloane's database of integer sequences.

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