Best Known (212−25, 212, s)-Nets in Base 4
(212−25, 212, 349534)-Net over F4 — Constructive and digital
Digital (187, 212, 349534)-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 (174, 199, 349525)-net over F4, using
- net defined by OOA [i] based on linear OOA(4199, 349525, F4, 25, 25) (dual of [(349525, 25), 8737926, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4199, 4194301, F4, 25) (dual of [4194301, 4194102, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(4199, 4194301, F4, 25) (dual of [4194301, 4194102, 26]-code), using
- net defined by OOA [i] based on linear OOA(4199, 349525, F4, 25, 25) (dual of [(349525, 25), 8737926, 26]-NRT-code), using
- digital (1, 13, 9)-net over F4, using
(212−25, 212, 1685247)-Net over F4 — Digital
Digital (187, 212, 1685247)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4212, 1685247, F4, 2, 25) (dual of [(1685247, 2), 3370282, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4212, 2097186, F4, 2, 25) (dual of [(2097186, 2), 4194160, 26]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4212, 4194372, F4, 25) (dual of [4194372, 4194160, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- linear OA(4199, 4194304, F4, 25) (dual of [4194304, 4194105, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(4144, 4194304, F4, 18) (dual of [4194304, 4194160, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(413, 68, F4, 6) (dual of [68, 55, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(413, 70, F4, 6) (dual of [70, 57, 7]-code), using
- construction X applied to Ce(24) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4212, 4194372, F4, 25) (dual of [4194372, 4194160, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(4212, 2097186, F4, 2, 25) (dual of [(2097186, 2), 4194160, 26]-NRT-code), using
(212−25, 212, large)-Net in Base 4 — Upper bound on s
There is no (187, 212, large)-net in base 4, because
- 23 times m-reduction [i] would yield (187, 189, large)-net in base 4, but