Spectral Orbits and Peak-to-Average Power Ratio of Boolean Functions with Respect to the {I, H, N}n Transform

We enumerate the inequivalent self-dual additive codes over GF(4) of blocklength n, thereby extending the sequence A090899 in The On-Line Encyclopedia of Integer Sequences from n = 9 to n = 12. These codes have a well-known interpretation as quantum codes. They can also be represented by graphs, where a simple graph operation generates the orbits of equivalent codes. We highlight the regularity and structure of some graphs that correspond to codes with high distance. The codes can also be interpreted as quadratic Boolean functions, where inequivalence takes on a spectral meaning. In this context we de ne PARIHN , peakto-average power ratio with respect to the fI;H;Ng transform set. We prove that PARIHN of a Boolean function is equivalent to the the size of the maximum independent set over the associated orbit of graphs. Finally we propose a construction technique to generate Boolean functions with low PARIHN and algebraic degree higher than 2. 1 Self-Dual Additive Codes over GF(4) A quantum error-correcting code with parameters [[n; k; d]] encodes k qubits in a highly entangled state of n qubits such that any error a ecting less than d qubits can be detected, and any error a ecting at most d 1 2 qubits can be corrected. A quantum code of the stabilizer type corresponds to a code C GF(4)n [1]. We denote GF(4) = f0; 1; !; !2g, where !2 = !+1. Conjugation in GF(4) is de ned by x = x2. The trace map, tr : GF(4) 7! GF(2), is de ned by tr(x) = x + x. The trace inner product of two vectors of length n over GF(4), u and v, is given by u v = Pn i=1 tr(uivi). Because of the structure of stabilizer codes, the corresponding code over GF(4), C, will be additive and satisfy u v = 0 for any two codewords u;v 2 C. This is equivalent to saying that the code must be self-orthogonal with respect to the trace inner product, i.e., C C?, where C? = fu 2 GF(4)n j u c = 0; 8c 2 Cg. We will only consider codes of the special case where the dimension k = 0. Zero-dimensional quantum codes can be understood as highly-entangled single quantum states which are robust to error. These codes map to additive codes 2 Lars Eirik Danielsen and Matthew G. Parker over GF(4) which are self-dual [2], C = C?. The number of inequivalent selfdual additive codes over GF(4) of blocklength n has been classi ed by Calderbank et al. [1] for n 5, by Hohn [3] for n 7, by Hein et al. [4] for n 7, and by Glynn et al. [5] for n 9. Moreover, Glynn has recently posted these results as sequence A090899 in The On-Line Encyclopedia of Integer Sequences [6]. We extend this sequence from n = 9 to n = 12 both for indecomposable and decomposable codes as shown in table 1. Table 2 shows the number of inequivalent indecomposable codes by distance. The distance, d, of a self-dual additive code over GF(4), C, is the smallest weight (i.e., number of nonzero components) of any nonzero codeword in C. A database of orbit representatives with information about orbit size, distance, and weight distribution is also available [7]. Table 1: Number of Inequivalent Indecomposable (in) and (Possibly) Decomposable (tn) Self-Dual Additive Codes Over GF(4) n 1 2 3 4 5 6 7 8 9 10 11 12 in 1 1 1 2 4 11 26 101 440 3,132 40,457 1,274,068 tn 1 2 3 6 11 26 59 182 675 3,990 45,144 1,323,363 Table 2:Number of Indecomposable Self-Dual Additive Codes Over GF(4) by Distance dnn 2 3 4 5 6 7 8 9 10 11 12 2 1 1 2 3 9 22 85 363 2,436 26,750 611,036 3 1 1 4 11 69 576 11,200 467,513 4 1 5 8 120 2,506 195,455 5 1 63 6 1 Total 1 1 2 4 11 26 101 440 3,132 40,457 1,274,068 2 Graphs, Boolean Functions, and LC-Equivalence A self-dual additive code over GF(4) corresponds to a graph state [4] if its generator matrix, G, can be written as G = + !I , where is a symmetric matrix over GF(2) with zeros on the diagonal. The matrix can be interpreted as the adjacency matrix of a simple undirected graph on n vertices. It has been shown by Schlingemann and Werner [8], Grassl et al. [9], Glynn [10], and Van den Nest et al. [11] that all stabilizer states can be transformed into an equivalent graph state. Thus all self-dual additive codes over GF(4) can be represented by graphs. These codes also have another interpretation as quadratic Boolean functions over n variables. A quadratic function, f , can be represented by an adjacency matrix, , where i;j = j;i = 1 if xixj occurs in f , and i;j = 0 otherwise. Spectral Orbits and PAR of Boolean Functions w.r.t. fI;H;Ng 3 Example 1. A self-dual additive code over GF(4) with parameters [[6; 0; 4]] is generated by the generator matrix 0 B B B B B B @ ! 0 0 1 1 1 0 ! 0 !2 1 ! 0 0 ! !2 ! 1 0 1 0 ! !2 1 0 0 1 ! 1 !2 1 !2 0 ! 0 0 1