Best Known (91, 91+18, s)-Nets in Base 4
(91, 91+18, 7284)-Net over F4 — Constructive and digital
Digital (91, 109, 7284)-net over F4, using
- net defined by OOA [i] based on linear OOA(4109, 7284, F4, 18, 18) (dual of [(7284, 18), 131003, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4109, 65556, F4, 18) (dual of [65556, 65447, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, 65557, F4, 18) (dual of [65557, 65448, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(43, 20, F4, 2) (dual of [20, 17, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4109, 65557, F4, 18) (dual of [65557, 65448, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(4109, 65556, F4, 18) (dual of [65556, 65447, 19]-code), using
(91, 91+18, 32778)-Net over F4 — Digital
Digital (91, 109, 32778)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4109, 32778, F4, 2, 18) (dual of [(32778, 2), 65447, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4109, 65556, F4, 18) (dual of [65556, 65447, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(4109, 65557, F4, 18) (dual of [65557, 65448, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- linear OA(4105, 65536, F4, 18) (dual of [65536, 65431, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(489, 65536, F4, 15) (dual of [65536, 65447, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(43, 20, F4, 2) (dual of [20, 17, 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
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(17) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4109, 65557, F4, 18) (dual of [65557, 65448, 19]-code), using
- OOA 2-folding [i] based on linear OA(4109, 65556, F4, 18) (dual of [65556, 65447, 19]-code), using
(91, 91+18, large)-Net in Base 4 — Upper bound on s
There is no (91, 109, large)-net in base 4, because
- 16 times m-reduction [i] would yield (91, 93, large)-net in base 4, but