Best Known (243−31, 243, s)-Nets in Base 4
(243−31, 243, 69909)-Net over F4 — Constructive and digital
Digital (212, 243, 69909)-net over F4, using
- 42 times duplication [i] based on digital (210, 241, 69909)-net over F4, using
- net defined by OOA [i] based on linear OOA(4241, 69909, F4, 31, 31) (dual of [(69909, 31), 2166938, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(4241, 1048636, F4, 31) (dual of [1048636, 1048395, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [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(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 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 Ce(30) ⊂ Ce(24) [i] based on
- OOA 15-folding and stacking with additional row [i] based on linear OA(4241, 1048636, F4, 31) (dual of [1048636, 1048395, 32]-code), using
- net defined by OOA [i] based on linear OOA(4241, 69909, F4, 31, 31) (dual of [(69909, 31), 2166938, 32]-NRT-code), using
(243−31, 243, 524319)-Net over F4 — Digital
Digital (212, 243, 524319)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(4243, 524319, F4, 2, 31) (dual of [(524319, 2), 1048395, 32]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4241, 524318, F4, 2, 31) (dual of [(524318, 2), 1048395, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(4241, 1048636, F4, 31) (dual of [1048636, 1048395, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(24) [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(4181, 1048576, F4, 25) (dual of [1048576, 1048395, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 410−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(410, 60, F4, 5) (dual of [60, 50, 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 Ce(30) ⊂ Ce(24) [i] based on
- OOA 2-folding [i] based on linear OA(4241, 1048636, F4, 31) (dual of [1048636, 1048395, 32]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(4241, 524318, F4, 2, 31) (dual of [(524318, 2), 1048395, 32]-NRT-code), using
(243−31, 243, large)-Net in Base 4 — Upper bound on s
There is no (212, 243, large)-net in base 4, because
- 29 times m-reduction [i] would yield (212, 214, large)-net in base 4, but