A Family of Alltop Functions that are EA-Inequivalent to the Cubic Function

Abstract

Sequences with optimal correlation properties are much sought after for applications in communication systems. In 1980, Alltop (IEEE Trans. Inf. Theory 26(3):350-354, 1980) described a set of sequences based on a cubic function and showed that these sequences were optimal with respect to the known bounds on auto and crosscorrelation. Subsequently these sequences were used to construct mutually unbiased bases (MUBs), a structure of importance in quantum information theory. The key feature of this cubic function is that its difference function is a planar function. Functions with planar difference functions have been called Alltop functions. This paper provides a new family of Alltop functions and establishes the use of Alltop functions for construction of sequence sets and MUBs.

Towards a classification of incomplete Gabor POVMs in $\mathbb{C}^d$

“…In fact we can show Proposition 4. (See [16,19]) If 1 ≤ a < d, d is an odd prime, and g is as in (11), then…”

Orbits of mutually unbiased bases

“…These order 3 unitaries appear in the orbit complete MUBs construction, too. As a consequence of the orbit complete MUBs lying in different Clifford orbits in dimension N = 3k + 1 [28], the Alltop fiducial vectors in Equation ( 16) must be left invariant by more Clifford unitaries than the one used in Equation (21). The additional unitaries that leave the fiducial invariant are the order 3 elements in the SIC problem.…”

“…Alltop's original paper was motivated by classical signal processing, but the functions were shown to lead to complete MUBs in prime and odd prime power dimensions [20]. Recently, a new family of Alltop functions was found [21].…”

Relationships Between CCZ and EA Equivalence Classes and Corresponding Code Invariants

“…A recent illustration of this appears in [15] where it is shown that for p ≥ 5 and m an integer that does not divide p m + 1, then the function f (x) = x p m +2 over Z m p is an Alltop function (that is, its differential functions are PN) which is EA-inequivalent to the Alltop function f (x) = x 3 , even though ∂f and ∂f are EA equivalent PN functions.…”

“…According to [19] there are 5 EA equivalent classes of differentially 4-uniform permutations, and as they all have different extended Walsh spectra, they each form a single CCZ equivalence class. On the other hand, using MAGMA [3] and checking all differentially 4-uniform permutations in S 15 , there are exactly 10 CCZ equivalence classes, given by the following permutations: [19]), σ 5 = (5, 6, 8) (7,11,14,10,12,13), σ 6 = (5, 6, 8)(7, 11, 14)(10, 12, 13) (= f 7 in [19]), σ 7 = (5, 6, 8)(7, 11, 13, 15)(9, 12, 10), σ 8 = (5, 6, 8)(7, 11, 13)(9, 12, 14, 10), σ 9 = (5, 6, 8) (7,11,13,10,9,12,14), σ 10 = (5, 6, 8)(7, 11) (9,12,10,13,15,14) . Table 4 corrects [19, Table 2], where the CCZ classes of σ 1 , σ 2 , σ 3 , σ 4 , σ 6 were claimed to exhaust the differentially 4-uniform classes of permutations fixing 0 over Z 4 2 .…”