Best Known (77, 77+18, s)-Nets in Base 4
(77, 77+18, 1822)-Net over F4 — Constructive and digital
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
(77, 77+18, 8200)-Net over F4 — Digital
Digital (77, 95, 8200)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(495, 8200, F4, 2, 18) (dual of [(8200, 2), 16305, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(495, 16400, F4, 18) (dual of [16400, 16305, 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
- OOA 2-folding [i] based on linear OA(495, 16400, F4, 18) (dual of [16400, 16305, 19]-code), using
(77, 77+18, 3131171)-Net in Base 4 — Upper bound on s
There is no (77, 95, 3131172)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1569 275954 362891 799646 009877 998665 690126 477687 860284 894844 > 495 [i]