Correlation immunity of the Boolean functions built from mutually orthogonal cellular automata: the exhaustive classification for local rules of diameter at most six

Definition fidelity against the source: every bipermutive local rule of diameter 3, 4, 5 induces a Latin square of order 2ᵈ⁻¹; the source's Figure 1 data (all eight coupled de Bruijn labels of rules 90 and 150 at d = 3, and their orthogonality) is reproduced exactly; and the number of orthogonal pairs of bipermutive CA of diameter 3, 4, 5 (4, 36, 852 unordered) reproduces Table 2 of the source's own reference Mariot-Formenti-Leporati 2017 (1, 9, 213) under that paper's stated convention of dividing by 8

Diameters 3 and 4: there is no 3-MOCA of diameter 3 (though 4 orthogonal pairs exist among the 4 bipermutive rules); at diameter 4 there are 16 bipermutive rules, 36 orthogonal pairs and exactly 16 sets of 3-MOCA, listed explicitly, which is 2 under the source's own divide-by-8 convention (equivalently 2 complementation orbits, all of size 8); and there is no 4-MOCA of diameter 4, so 3 is the maximum family size at that diameter

Diameter 4 correlation immunity: every one of the 16 sets of 3-MOCA has 64 pairwise distinct rows, so its Boolean function has n = 9 variables and Hamming weight 64, and its correlation immunity order is exactly 3 – never 2, and never 4

Diameter 5, k = 3: among the 256 bipermutive rules there are 852 orthogonal pairs and exactly 264 sets of 3-MOCA, each giving a 12-variable Boolean function of Hamming weight 256 with 256 pairwise distinct rows; 264 is 33 under the source's own divide-by-8 convention (equivalently 33 complementation orbits, all of size 8), where Table 1 of the source prints 36

Diameter 5, k = 3, the exact correlation immunity distribution: of the 264 sets of 3-MOCA exactly 192 give a function of correlation immunity order exactly 3 and exactly 72 give order exactly 4, and no other order occurs; under the source's divide-by-8 convention that is 24 and 9, where Table 1 of the published source prints 27 and 9 (and arXiv v1 prints 27 and 6)

The source's open question, answered exhaustively at diameter 5: no 3-MOCA of diameter 5 gives a function that is only second-order correlation immune – all 264 have order at least 3, so Theorem 1's bound is attained by nothing; the maximum order attained is 4, and both 3 and 4 occur

Diameter 5 beyond k = 3, a cell the source never enters: there are exactly 224 sets of 4-MOCA and exactly 96 sets of 5-MOCA of diameter 5 and no 6-MOCA, so the maximum family size is 5; every one of the 320 gives a Boolean function of Hamming weight 256 on n = 16 respectively n = 20 variables whose correlation immunity order is exactly 3; and every one of the 852 sets of 2-MOCA has all 256 rows distinct, so its support is the whole of F₂⁸ and it defines the constant function, whose order is the maximum 8

Diameter 6, certified witnesses for the column the arXiv table's caption promises and its body omits: named 3-MOCA families of diameter 6 with correlation immunity order exactly 5, exactly 4 and exactly 3 (so the maximum order at d = 6 is at least 5, strictly above the maximum 4 at d = 5); a 4-MOCA of diameter 6 of order exactly 4 (impossible at d = 5); and an 8-MOCA of diameter 6, giving a 40-variable third-order correlation immune function of Hamming weight 1024

The complete diameter-6 classification, computed but not certified: 65536 bipermutive rules, 266740 orthogonal pairs (66685 under the source's divide-by-8 convention, reproducing the published d = 6 entry of Mariot-Formenti-Leporati Table 2), 5808 sets of 3-MOCA (726 reduced) on n = 15 variables of weight 1024 with correlation immunity orders 3, 4, 5 in 4576, 960, 272 families (572, 120, 34 reduced), and 6160, 11392, 13504, 8960, 2560 families at k = 4, 5, 6, 7, 8 with no 9-MOCA – every one of order exactly 3 except 416 of the 4-MOCA, which have order 4. The minimum order over every family of every k at diameter 6 is 3

Negative controls and definition fidelity: the linear function x₁ xor... xor x₆ is recovered as correlation immune of order exactly 5 by both routes; the indicator of a single point of F₂⁴ is rejected as not correlation immune at all; the fast Walsh-Hadamard transform agrees with the source's Eq. (3) written out over the whole input space in all 512 coefficients of a d = 4 family; orthogonality is neither always true nor always false; the 16 rule encodings give 16 distinct Latin squares; dropping pairwise orthogonality collapses the strength to 1 while leaving the rows distinct; complementing the local rules permutes the diameter-4 families and preserves their correlation immunity order; and the too-large claim 'order at least 4 at d = 4' and the too-small claim 'order at most 2 at d = 4' are both refuted

The Min w_H column of the source's Table 1 is wrong in its last row: for n = 12 variables and correlation immunity order 4 it prints 24, which is a duplicate of the order-3 entry, whereas the minimum Hamming weight of a 4th-order correlation immune Boolean function in 12 variables is 128; consequently the source's own conclusion that 'for n = 12 variables the gap is even greater' overstates the order-4 gap by a factor of more than 5 (the true gap is 256/128 = 2). Separately, the printed 20 for (n, t) = (9, 3) cannot be the minimum weight, because the weight of a t-th order correlation immune function is divisible by 2ᵗ and the Rao bound already forces at least 18, hence at least 24