Best Known (206−26, 206, s)-Nets in Base 4
(206−26, 206, 80670)-Net over F4 — Constructive and digital
Digital (180, 206, 80670)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 15, 10)-net over F4, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 2 and N(F) ≥ 10, using
- net from sequence [i] based on digital (2, 9)-sequence over F4, using
- digital (165, 191, 80660)-net over F4, using
- net defined by OOA [i] based on linear OOA(4191, 80660, F4, 26, 26) (dual of [(80660, 26), 2096969, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(4191, 1048580, F4, 26) (dual of [1048580, 1048389, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(40, 10, F4, 0) (dual of [10, 10, 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
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(4191, 1048586, F4, 26) (dual of [1048586, 1048395, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(4191, 1048580, F4, 26) (dual of [1048580, 1048389, 27]-code), using
- net defined by OOA [i] based on linear OOA(4191, 80660, F4, 26, 26) (dual of [(80660, 26), 2096969, 27]-NRT-code), using
- digital (2, 15, 10)-net over F4, using
(206−26, 206, 524325)-Net over F4 — Digital
Digital (180, 206, 524325)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4206, 524325, F4, 2, 26) (dual of [(524325, 2), 1048444, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4206, 1048650, F4, 26) (dual of [1048650, 1048444, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(4206, 1048651, F4, 26) (dual of [1048651, 1048445, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- linear OA(4191, 1048576, F4, 26) (dual of [1048576, 1048385, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(4131, 1048576, F4, 18) (dual of [1048576, 1048445, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(415, 75, F4, 7) (dual of [75, 60, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(415, 85, F4, 7) (dual of [85, 70, 8]-code), using
- construction X applied to Ce(25) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4206, 1048651, F4, 26) (dual of [1048651, 1048445, 27]-code), using
- OOA 2-folding [i] based on linear OA(4206, 1048650, F4, 26) (dual of [1048650, 1048444, 27]-code), using
(206−26, 206, large)-Net in Base 4 — Upper bound on s
There is no (180, 206, large)-net in base 4, because
- 24 times m-reduction [i] would yield (180, 182, large)-net in base 4, but