Best Known (181−24, 181, s)-Nets in Base 4
(181−24, 181, 87382)-Net over F4 — Constructive and digital
Digital (157, 181, 87382)-net over F4, using
- net defined by OOA [i] based on linear OOA(4181, 87382, F4, 24, 24) (dual of [(87382, 24), 2096987, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(4181, 1048584, F4, 24) (dual of [1048584, 1048403, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 1048586, F4, 24) (dual of [1048586, 1048405, 25]-code), using
- 1 times truncation [i] based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4181, 1048586, F4, 24) (dual of [1048586, 1048405, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(4181, 1048584, F4, 24) (dual of [1048584, 1048403, 25]-code), using
(181−24, 181, 391920)-Net over F4 — Digital
Digital (157, 181, 391920)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4181, 391920, F4, 2, 24) (dual of [(391920, 2), 783659, 25]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4181, 524293, F4, 2, 24) (dual of [(524293, 2), 1048405, 25]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4181, 1048586, F4, 24) (dual of [1048586, 1048405, 25]-code), using
- 1 times truncation [i] based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- linear OA(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4171, 1048576, F4, 23) (dual of [1048576, 1048405, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(41, 11, F4, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(24) ⊂ Ce(22) [i] based on
- 1 times truncation [i] based on linear OA(4182, 1048587, F4, 25) (dual of [1048587, 1048405, 26]-code), using
- OOA 2-folding [i] based on linear OA(4181, 1048586, F4, 24) (dual of [1048586, 1048405, 25]-code), using
- discarding factors / shortening the dual code based on linear OOA(4181, 524293, F4, 2, 24) (dual of [(524293, 2), 1048405, 25]-NRT-code), using
(181−24, 181, large)-Net in Base 4 — Upper bound on s
There is no (157, 181, large)-net in base 4, because
- 22 times m-reduction [i] would yield (157, 159, large)-net in base 4, but