Best Known (210, 210+41, s)-Nets in Base 4
(210, 210+41, 3278)-Net over F4 — Constructive and digital
Digital (210, 251, 3278)-net over F4, using
- 45 times duplication [i] based on digital (205, 246, 3278)-net over F4, using
- net defined by OOA [i] based on linear OOA(4246, 3278, F4, 41, 41) (dual of [(3278, 41), 134152, 42]-NRT-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4246, 65561, F4, 41) (dual of [65561, 65315, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4246, 65565, F4, 41) (dual of [65565, 65319, 42]-code), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- linear OA(4241, 65536, F4, 41) (dual of [65536, 65295, 42]-code), using an extension Ce(40) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,40], and designed minimum distance d ≥ |I|+1 = 41 [i]
- linear OA(4217, 65536, F4, 37) (dual of [65536, 65319, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(45, 29, F4, 3) (dual of [29, 24, 4]-code or 29-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(40) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(4246, 65565, F4, 41) (dual of [65565, 65319, 42]-code), using
- OOA 20-folding and stacking with additional row [i] based on linear OA(4246, 65561, F4, 41) (dual of [65561, 65315, 42]-code), using
- net defined by OOA [i] based on linear OOA(4246, 3278, F4, 41, 41) (dual of [(3278, 41), 134152, 42]-NRT-code), using
(210, 210+41, 37093)-Net over F4 — Digital
Digital (210, 251, 37093)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4251, 37093, F4, 41) (dual of [37093, 36842, 42]-code), using
- discarding factors / shortening the dual code based on linear OA(4251, 65579, F4, 41) (dual of [65579, 65328, 42]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(410, 42, F4, 5) (dual of [42, 32, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 43−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(410, 63, F4, 5) (dual of [63, 53, 6]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4251, 65579, F4, 41) (dual of [65579, 65328, 42]-code), using
(210, 210+41, large)-Net in Base 4 — Upper bound on s
There is no (210, 251, large)-net in base 4, because
- 39 times m-reduction [i] would yield (210, 212, large)-net in base 4, but