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doi: 10.3934/amc.2021011
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## The weight recursions for the 2-rotation symmetric quartic Boolean functions

 1 Department of Mathematics, University at Buffalo, 244 Mathematics Bldg., Buffalo, NY 14260 2 Department of Military Operations Research, Korea Army Academy at YeongCheon, 135-9, Hoguk-ro, Gogyeong-myeon, Yeongcheon-si, Gyeongsangbuk-do, Republic of Korea, 38900

Received  October 2020 Revised  February 2021 Early access May 2021

A Boolean function in $n$ variables is 2-rotation symmetric if it is invariant under even powers of $\rho(x_1, \ldots, x_n) = (x_2, \ldots, x_n, x_1)$, but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of $\rho^2$ to a single monomial. If the quartic MRS 2-function in $2n$ variables has a monomial $x_1 x_q x_r x_s$, then we use the notation ${2-}(1,q,r,s)_{2n}$ for the function. A detailed theory of equivalence of quartic MRS 2-functions in $2n$ variables was given in a $2020$ paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called $mf1$ and $mf2$ in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions ${2-}(1,q,r,s)_{2n}$ (with $q < r < s,$ say), $n = s, s+1, \ldots$ can be shown to satisfy. This problem was solved for the $mf1$ case only in the $2020$ paper. Using new ideas about "short" functions, Cusick and Cheon found formulas for the $mf2$ weights in a $2021$ sequel to the $2020$ paper. In this paper the actual recursions for the weights in the $mf2$ case are determined.

Citation: Thomas W. Cusick, Younhwan Cheon. The weight recursions for the 2-rotation symmetric quartic Boolean functions. Advances in Mathematics of Communications, doi: 10.3934/amc.2021011
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show all references

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The recursion polynomial for $f$ corresponding to some $\mu$ values
 $\mu$ $p(x)$ $2$ $x^3-4x^2-8x+32=(x-4)(x^2-8)$ $4$ $x^5-4x^4-64x+256=(x-4)(x^4-64)$ $6$ $x^7-4x^6-512x+2048=(x-4)(x^6-512)$ $8$ $x^9-4x^8-4096x+16384=(x-4)(x^8-4096)$ $t-1$($t$ is odd) $x^t -4x^{t-1} -2^{(3t-3)/2}x + 2^{(3t+1)/2}= (x-4)(x^{t-1}-2^{(3t-3)/2})$
 $\mu$ $p(x)$ $2$ $x^3-4x^2-8x+32=(x-4)(x^2-8)$ $4$ $x^5-4x^4-64x+256=(x-4)(x^4-64)$ $6$ $x^7-4x^6-512x+2048=(x-4)(x^6-512)$ $8$ $x^9-4x^8-4096x+16384=(x-4)(x^8-4096)$ $t-1$($t$ is odd) $x^t -4x^{t-1} -2^{(3t-3)/2}x + 2^{(3t+1)/2}= (x-4)(x^{t-1}-2^{(3t-3)/2})$
List of $\chi:F_\chi(x)$
 $\chi$ $F_\chi(x)$ 2 $(x-4)(x^2-2x-6)$ 4 $(x-4)(x^2-2x-6)(x^2+6)$ 6 $(x-4)(x^2-2x-6)(x^6+12x^3-216)$ 8 $(x-4)(x^2-2x-6)(x^2+6)(x^2-6)(x^8+96x^4+1296)$ 10 $(x-4)(x^2-2x-6)(x^{10}-72x^5-7776)(x^{10}+264x^5-7776)$ 12 $(x-4)(x^2-2x-6)(x^2+6)(x^4-6x^2+36)(x^6-12x^3-216)$ $(x^6+12x^3-216)(x^{12}+1104x^6+46656)$ 14 $(x-4)(x^2-2x-6)(x^{14}-1584x^7-2779936)(x^{14}+432x^7-2779936)$ $(x^{14}+3792x^7-2779936)$ 16 $(x-4)(x^2-2x-6)(x^2+6)(x^2-6)(x^4+36)(x^8-96x^4+1296)$ $(x^8+96x^4+1296)(x^{16}+3456x^8+1679616)(x^{16}+14208x^8+1679616)$
 $\chi$ $F_\chi(x)$ 2 $(x-4)(x^2-2x-6)$ 4 $(x-4)(x^2-2x-6)(x^2+6)$ 6 $(x-4)(x^2-2x-6)(x^6+12x^3-216)$ 8 $(x-4)(x^2-2x-6)(x^2+6)(x^2-6)(x^8+96x^4+1296)$ 10 $(x-4)(x^2-2x-6)(x^{10}-72x^5-7776)(x^{10}+264x^5-7776)$ 12 $(x-4)(x^2-2x-6)(x^2+6)(x^4-6x^2+36)(x^6-12x^3-216)$ $(x^6+12x^3-216)(x^{12}+1104x^6+46656)$ 14 $(x-4)(x^2-2x-6)(x^{14}-1584x^7-2779936)(x^{14}+432x^7-2779936)$ $(x^{14}+3792x^7-2779936)$ 16 $(x-4)(x^2-2x-6)(x^2+6)(x^2-6)(x^4+36)(x^8-96x^4+1296)$ $(x^8+96x^4+1296)(x^{16}+3456x^8+1679616)(x^{16}+14208x^8+1679616)$
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