Best Known (176, 176+29, s)-Nets in Base 4
(176, 176+29, 18733)-Net over F4 — Constructive and digital
Digital (176, 205, 18733)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 15, 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 (161, 190, 18724)-net over F4, using
- net defined by OOA [i] based on linear OOA(4190, 18724, F4, 29, 29) (dual of [(18724, 29), 542806, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4190, 262137, F4, 29) (dual of [262137, 261947, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- discarding factors / shortening the dual code based on linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(4190, 262137, F4, 29) (dual of [262137, 261947, 30]-code), using
- net defined by OOA [i] based on linear OOA(4190, 18724, F4, 29, 29) (dual of [(18724, 29), 542806, 30]-NRT-code), using
- digital (1, 15, 9)-net over F4, using
(176, 176+29, 131106)-Net over F4 — Digital
Digital (176, 205, 131106)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4205, 131106, F4, 2, 29) (dual of [(131106, 2), 262007, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4205, 262212, F4, 29) (dual of [262212, 262007, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(4205, 262213, F4, 29) (dual of [262213, 262008, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(20) [i] based on
- linear OA(4190, 262144, F4, 29) (dual of [262144, 261954, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(4136, 262144, F4, 21) (dual of [262144, 262008, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(415, 69, F4, 7) (dual of [69, 54, 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(28) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(4205, 262213, F4, 29) (dual of [262213, 262008, 30]-code), using
- OOA 2-folding [i] based on linear OA(4205, 262212, F4, 29) (dual of [262212, 262007, 30]-code), using
(176, 176+29, large)-Net in Base 4 — Upper bound on s
There is no (176, 205, large)-net in base 4, because
- 27 times m-reduction [i] would yield (176, 178, large)-net in base 4, but