Best Known (202, 202+31, s)-Nets in Base 4
(202, 202+31, 69906)-Net over F4 — Constructive and digital
Digital (202, 233, 69906)-net over F4, using
- 41 times duplication [i] based on digital (201, 232, 69906)-net over F4, using
- net defined by OOA [i] based on linear OOA(4232, 69906, F4, 31, 31) (dual of [(69906, 31), 2166854, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4232, 1048591, F4, 31) (dual of [1048591, 1048359, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4232, 1048597, F4, 31) (dual of [1048597, 1048365, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4232, 1048597, F4, 31) (dual of [1048597, 1048365, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4232, 1048591, F4, 31) (dual of [1048591, 1048359, 32]-code), using
- net defined by OOA [i] based on linear OOA(4232, 69906, F4, 31, 31) (dual of [(69906, 31), 2166854, 32]-NRT-code), using
(202, 202+31, 349532)-Net over F4 — Digital
Digital (202, 233, 349532)-net over F4, using
- 41 times duplication [i] based on digital (201, 232, 349532)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4232, 349532, F4, 3, 31) (dual of [(349532, 3), 1048364, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(4232, 1048596, F4, 31) (dual of [1048596, 1048364, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(4232, 1048597, F4, 31) (dual of [1048597, 1048365, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(4231, 1048576, F4, 31) (dual of [1048576, 1048345, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(4211, 1048576, F4, 29) (dual of [1048576, 1048365, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(41, 21, F4, 1) (dual of [21, 20, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(41, s, F4, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(4232, 1048597, F4, 31) (dual of [1048597, 1048365, 32]-code), using
- OOA 3-folding [i] based on linear OA(4232, 1048596, F4, 31) (dual of [1048596, 1048364, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4232, 349532, F4, 3, 31) (dual of [(349532, 3), 1048364, 32]-NRT-code), using
(202, 202+31, large)-Net in Base 4 — Upper bound on s
There is no (202, 233, large)-net in base 4, because
- 29 times m-reduction [i] would yield (202, 204, large)-net in base 4, but