Best Known (78, 78+18, s)-Nets in Base 4
(78, 78+18, 1822)-Net over F4 — Constructive and digital
Digital (78, 96, 1822)-net over F4, using
- 41 times duplication [i] based on digital (77, 95, 1822)-net over F4, using
- net defined by OOA [i] based on linear OOA(495, 1822, F4, 18, 18) (dual of [(1822, 18), 32701, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(495, 16398, F4, 18) (dual of [16398, 16303, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(495, 16401, F4, 18) (dual of [16401, 16306, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(14) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [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(17) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(495, 16401, F4, 18) (dual of [16401, 16306, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(495, 16398, F4, 18) (dual of [16398, 16303, 19]-code), using
- net defined by OOA [i] based on linear OOA(495, 1822, F4, 18, 18) (dual of [(1822, 18), 32701, 19]-NRT-code), using
(78, 78+18, 8502)-Net over F4 — Digital
Digital (78, 96, 8502)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(496, 8502, F4, 18) (dual of [8502, 8406, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(496, 16403, F4, 18) (dual of [16403, 16307, 19]-code), using
- construction XX applied to Ce(17) ⊂ Ce(14) ⊂ Ce(13) [i] based on
- linear OA(492, 16384, F4, 18) (dual of [16384, 16292, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(478, 16384, F4, 15) (dual of [16384, 16306, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(471, 16384, F4, 14) (dual of [16384, 16313, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(43, 18, F4, 2) (dual of [18, 15, 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(496, 16403, F4, 18) (dual of [16403, 16307, 19]-code), using
(78, 78+18, 3652604)-Net in Base 4 — Upper bound on s
There is no (78, 96, 3652605)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 6277 115701 954117 714068 833035 597283 711495 127685 492652 035744 > 496 [i]