Best Known (132, 132+18, s)-Nets in Base 4
(132, 132+18, 466038)-Net over F4 — Constructive and digital
Digital (132, 150, 466038)-net over F4, using
- 41 times duplication [i] based on digital (131, 149, 466038)-net over F4, using
- net defined by OOA [i] based on linear OOA(4149, 466038, F4, 18, 18) (dual of [(466038, 18), 8388535, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(4149, 4194342, F4, 18) (dual of [4194342, 4194193, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- 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(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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(17) ⊂ Ce(13) [i] based on
- OA 9-folding and stacking [i] based on linear OA(4149, 4194342, F4, 18) (dual of [4194342, 4194193, 19]-code), using
- net defined by OOA [i] based on linear OOA(4149, 466038, F4, 18, 18) (dual of [(466038, 18), 8388535, 19]-NRT-code), using
(132, 132+18, 1866242)-Net over F4 — Digital
Digital (132, 150, 1866242)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4150, 1866242, F4, 2, 18) (dual of [(1866242, 2), 3732334, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4150, 2097171, F4, 2, 18) (dual of [(2097171, 2), 4194192, 19]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4149, 2097171, F4, 2, 18) (dual of [(2097171, 2), 4194193, 19]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4149, 4194342, F4, 18) (dual of [4194342, 4194193, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(13) [i] based on
- 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(4111, 4194304, F4, 14) (dual of [4194304, 4194193, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(45, 38, F4, 3) (dual of [38, 33, 4]-code or 38-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(17) ⊂ Ce(13) [i] based on
- OOA 2-folding [i] based on linear OA(4149, 4194342, F4, 18) (dual of [4194342, 4194193, 19]-code), using
- 41 times duplication [i] based on linear OOA(4149, 2097171, F4, 2, 18) (dual of [(2097171, 2), 4194193, 19]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4150, 2097171, F4, 2, 18) (dual of [(2097171, 2), 4194192, 19]-NRT-code), using
(132, 132+18, large)-Net in Base 4 — Upper bound on s
There is no (132, 150, large)-net in base 4, because
- 16 times m-reduction [i] would yield (132, 134, large)-net in base 4, but