Best Known (151, 151+25, s)-Nets in Base 4
(151, 151+25, 21854)-Net over F4 — Constructive and digital
Digital (151, 176, 21854)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 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 (138, 163, 21845)-net over F4, using
- net defined by OOA [i] based on linear OOA(4163, 21845, F4, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4163, 262141, F4, 25) (dual of [262141, 261978, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4163, 262141, F4, 25) (dual of [262141, 261978, 26]-code), using
- net defined by OOA [i] based on linear OOA(4163, 21845, F4, 25, 25) (dual of [(21845, 25), 545962, 26]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(151, 151+25, 131101)-Net over F4 — Digital
Digital (151, 176, 131101)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4176, 131101, F4, 2, 25) (dual of [(131101, 2), 262026, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4176, 262202, F4, 25) (dual of [262202, 262026, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4163, 262144, F4, 25) (dual of [262144, 261981, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4118, 262144, F4, 18) (dual of [262144, 262026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(413, 58, F4, 6) (dual of [58, 45, 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(24) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4176, 262202, F4, 25) (dual of [262202, 262026, 26]-code), using
(151, 151+25, large)-Net in Base 4 — Upper bound on s
There is no (151, 176, large)-net in base 4, because
- 23 times m-reduction [i] would yield (151, 153, large)-net in base 4, but