Best Known (161, 161+22, s)-Nets in Base 4
(161, 161+22, 381303)-Net over F4 — Constructive and digital
Digital (161, 183, 381303)-net over F4, using
- 41 times duplication [i] based on digital (160, 182, 381303)-net over F4, using
- net defined by OOA [i] based on linear OOA(4182, 381303, F4, 22, 22) (dual of [(381303, 22), 8388484, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(4182, 4194333, F4, 22) (dual of [4194333, 4194151, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(4182, 4194342, F4, 22) (dual of [4194342, 4194160, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(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(21) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(4182, 4194342, F4, 22) (dual of [4194342, 4194160, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(4182, 4194333, F4, 22) (dual of [4194333, 4194151, 23]-code), using
- net defined by OOA [i] based on linear OOA(4182, 381303, F4, 22, 22) (dual of [(381303, 22), 8388484, 23]-NRT-code), using
(161, 161+22, 1437154)-Net over F4 — Digital
Digital (161, 183, 1437154)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4183, 1437154, F4, 2, 22) (dual of [(1437154, 2), 2874125, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4183, 2097171, F4, 2, 22) (dual of [(2097171, 2), 4194159, 23]-NRT-code), using
- 41 times duplication [i] based on linear OOA(4182, 2097171, F4, 2, 22) (dual of [(2097171, 2), 4194160, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4182, 4194342, F4, 22) (dual of [4194342, 4194160, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(17) [i] based on
- linear OA(4177, 4194304, F4, 22) (dual of [4194304, 4194127, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 4194303 = 411−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(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(21) ⊂ Ce(17) [i] based on
- OOA 2-folding [i] based on linear OA(4182, 4194342, F4, 22) (dual of [4194342, 4194160, 23]-code), using
- 41 times duplication [i] based on linear OOA(4182, 2097171, F4, 2, 22) (dual of [(2097171, 2), 4194160, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(4183, 2097171, F4, 2, 22) (dual of [(2097171, 2), 4194159, 23]-NRT-code), using
(161, 161+22, large)-Net in Base 4 — Upper bound on s
There is no (161, 183, large)-net in base 4, because
- 20 times m-reduction [i] would yield (161, 163, large)-net in base 4, but