Best Known (193−30, 193, s)-Nets in Base 4
(193−30, 193, 4378)-Net over F4 — Constructive and digital
Digital (163, 193, 4378)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 16, 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 (147, 177, 4369)-net over F4, using
- net defined by OOA [i] based on linear OOA(4177, 4369, F4, 30, 30) (dual of [(4369, 30), 130893, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(4177, 65535, F4, 30) (dual of [65535, 65358, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using
- an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- discarding factors / shortening the dual code based on linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(4177, 65535, F4, 30) (dual of [65535, 65358, 31]-code), using
- net defined by OOA [i] based on linear OOA(4177, 4369, F4, 30, 30) (dual of [(4369, 30), 130893, 31]-NRT-code), using
- digital (1, 16, 9)-net over F4, using
(193−30, 193, 50595)-Net over F4 — Digital
Digital (163, 193, 50595)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4193, 50595, F4, 30) (dual of [50595, 50402, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(4193, 65600, F4, 30) (dual of [65600, 65407, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(21) [i] based on
- linear OA(4177, 65536, F4, 30) (dual of [65536, 65359, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(4129, 65536, F4, 22) (dual of [65536, 65407, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(416, 64, F4, 7) (dual of [64, 48, 8]-code), using
- an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- construction X applied to Ce(29) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(4193, 65600, F4, 30) (dual of [65600, 65407, 31]-code), using
(193−30, 193, large)-Net in Base 4 — Upper bound on s
There is no (163, 193, large)-net in base 4, because
- 28 times m-reduction [i] would yield (163, 165, large)-net in base 4, but