Best Known (117−22, 117, s)-Nets in Base 4
(117−22, 117, 1491)-Net over F4 — Constructive and digital
Digital (95, 117, 1491)-net over F4, using
- 41 times duplication [i] based on digital (94, 116, 1491)-net over F4, using
- net defined by OOA [i] based on linear OOA(4116, 1491, F4, 22, 22) (dual of [(1491, 22), 32686, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4116, 16401, F4, 22) (dual of [16401, 16285, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(21) ⊂ Ce(18) [i] based on
- OA 11-folding and stacking [i] based on linear OA(4116, 16401, F4, 22) (dual of [16401, 16285, 23]-code), using
- net defined by OOA [i] based on linear OOA(4116, 1491, F4, 22, 22) (dual of [(1491, 22), 32686, 23]-NRT-code), using
(117−22, 117, 8577)-Net over F4 — Digital
Digital (95, 117, 8577)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4117, 8577, F4, 22) (dual of [8577, 8460, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4117, 16403, F4, 22) (dual of [16403, 16286, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- linear OA(4113, 16384, F4, 22) (dual of [16384, 16271, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(499, 16384, F4, 19) (dual of [16384, 16285, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 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(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(21) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4117, 16403, F4, 22) (dual of [16403, 16286, 23]-code), using
(117−22, 117, 4145920)-Net in Base 4 — Upper bound on s
There is no (95, 117, 4145921)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 27607 006452 386249 344344 240402 561359 723336 155705 541993 378544 502443 122944 > 4117 [i]