Best Known (98, 98+21, s)-Nets in Base 4
(98, 98+21, 1652)-Net over F4 — Constructive and digital
Digital (98, 119, 1652)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 14)-net over F4, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 3 and N(F) ≥ 14, using
- net from sequence [i] based on digital (3, 13)-sequence over F4, using
- digital (85, 106, 1638)-net over F4, using
- net defined by OOA [i] based on linear OOA(4106, 1638, F4, 21, 21) (dual of [(1638, 21), 34292, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4106, 16381, F4, 21) (dual of [16381, 16275, 22]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(4106, 16384, F4, 21) (dual of [16384, 16278, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(4106, 16381, F4, 21) (dual of [16381, 16275, 22]-code), using
- net defined by OOA [i] based on linear OOA(4106, 1638, F4, 21, 21) (dual of [(1638, 21), 34292, 22]-NRT-code), using
- digital (3, 13, 14)-net over F4, using
(98, 98+21, 14480)-Net over F4 — Digital
Digital (98, 119, 14480)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4119, 14480, F4, 21) (dual of [14480, 14361, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(4119, 16432, F4, 21) (dual of [16432, 16313, 22]-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(413, 48, F4, 6) (dual of [48, 35, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(413, 63, F4, 6) (dual of [63, 50, 7]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(4119, 16432, F4, 21) (dual of [16432, 16313, 22]-code), using
(98, 98+21, large)-Net in Base 4 — Upper bound on s
There is no (98, 119, large)-net in base 4, because
- 19 times m-reduction [i] would yield (98, 100, large)-net in base 4, but