Best Known (116−20, 116, s)-Nets in Base 4
(116−20, 116, 1647)-Net over F4 — Constructive and digital
Digital (96, 116, 1647)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 9)-net over F4, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- the Hermitian function field over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 1 and N(F) ≥ 9, using
- net from sequence [i] based on digital (1, 8)-sequence over F4, using
- digital (85, 105, 1638)-net over F4, using
- net defined by OOA [i] based on linear OOA(4105, 1638, F4, 20, 20) (dual of [(1638, 20), 32655, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(4105, 16380, F4, 20) (dual of [16380, 16275, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 16383, F4, 20) (dual of [16383, 16278, 21]-code), using
- 1 times truncation [i] based on linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 1 times truncation [i] based on linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4105, 16383, F4, 20) (dual of [16383, 16278, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(4105, 16380, F4, 20) (dual of [16380, 16275, 21]-code), using
- net defined by OOA [i] based on linear OOA(4105, 1638, F4, 20, 20) (dual of [(1638, 20), 32655, 21]-NRT-code), using
- digital (1, 11, 9)-net over F4, using
(116−20, 116, 16430)-Net over F4 — Digital
Digital (96, 116, 16430)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4116, 16430, F4, 20) (dual of [16430, 16314, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(4115, 16428, F4, 20) (dual of [16428, 16313, 21]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(49, 44, F4, 5) (dual of [44, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- a “DaH†code from Brouwer’s database [i]
- discarding factors / shortening the dual code based on linear OA(49, 51, F4, 5) (dual of [51, 42, 6]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(4115, 16429, F4, 19) (dual of [16429, 16314, 20]-code), using Gilbert–Varšamov bound and bm = 4115 > Vbs−1(k−1) = 455 489831 845592 430341 691130 748592 684562 362404 211053 951473 224088 553614 [i]
- linear OA(40, 1, F4, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(4115, 16428, F4, 20) (dual of [16428, 16313, 21]-code), using
- construction X with Varšamov bound [i] based on
(116−20, 116, large)-Net in Base 4 — Upper bound on s
There is no (96, 116, large)-net in base 4, because
- 18 times m-reduction [i] would yield (96, 98, large)-net in base 4, but