Best Known (26, 26+6, s)-Nets in Base 4
(26, 26+6, 5468)-Net over F4 — Constructive and digital
Digital (26, 32, 5468)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (0, 3, 5)-net over F4, using
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 0 and N(F) ≥ 5, using
- the rational function field F4(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 4)-sequence over F4, using
- digital (23, 29, 5463)-net over F4, using
- net defined by OOA [i] based on linear OOA(429, 5463, F4, 6, 6) (dual of [(5463, 6), 32749, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(429, 16389, F4, 6) (dual of [16389, 16360, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(429, 16391, F4, 6) (dual of [16391, 16362, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(422, 16384, F4, 5) (dual of [16384, 16362, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(40, 7, F4, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(429, 16391, F4, 6) (dual of [16391, 16362, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(429, 16389, F4, 6) (dual of [16389, 16360, 7]-code), using
- net defined by OOA [i] based on linear OOA(429, 5463, F4, 6, 6) (dual of [(5463, 6), 32749, 7]-NRT-code), using
- digital (0, 3, 5)-net over F4, using
(26, 26+6, 16401)-Net over F4 — Digital
Digital (26, 32, 16401)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(432, 16401, F4, 6) (dual of [16401, 16369, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(429, 16384, F4, 6) (dual of [16384, 16355, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(415, 16384, F4, 3) (dual of [16384, 16369, 4]-code or 16384-cap in PG(14,4)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(43, 17, F4, 2) (dual of [17, 14, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- Hamming code H(3,4) [i]
- discarding factors / shortening the dual code based on linear OA(43, 21, F4, 2) (dual of [21, 18, 3]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
(26, 26+6, 1600424)-Net in Base 4 — Upper bound on s
There is no (26, 32, 1600425)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 18 446761 058620 735576 > 432 [i]