Best Known (75−14, 75, s)-Nets in Base 4
(75−14, 75, 2343)-Net over F4 — Constructive and digital
Digital (61, 75, 2343)-net over F4, using
- 41 times duplication [i] based on digital (60, 74, 2343)-net over F4, using
- net defined by OOA [i] based on linear OOA(474, 2343, F4, 14, 14) (dual of [(2343, 14), 32728, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(474, 16401, F4, 14) (dual of [16401, 16327, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(10) [i] based on
- 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(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(13) ⊂ Ce(10) [i] based on
- OA 7-folding and stacking [i] based on linear OA(474, 16401, F4, 14) (dual of [16401, 16327, 15]-code), using
- net defined by OOA [i] based on linear OOA(474, 2343, F4, 14, 14) (dual of [(2343, 14), 32728, 15]-NRT-code), using
(75−14, 75, 9089)-Net over F4 — Digital
Digital (61, 75, 9089)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(475, 9089, F4, 14) (dual of [9089, 9014, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(475, 16403, F4, 14) (dual of [16403, 16328, 15]-code), using
- construction XX applied to Ce(13) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- 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(457, 16384, F4, 11) (dual of [16384, 16327, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(450, 16384, F4, 10) (dual of [16384, 16334, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(13) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(475, 16403, F4, 14) (dual of [16403, 16328, 15]-code), using
(75−14, 75, 3180090)-Net in Base 4 — Upper bound on s
There is no (61, 75, 3180091)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 1427 250419 039525 771233 613800 803830 010501 883648 > 475 [i]