Best Known (243−39, 243, s)-Nets in Base 4
(243−39, 243, 3451)-Net over F4 — Constructive and digital
Digital (204, 243, 3451)-net over F4, using
- 43 times duplication [i] based on digital (201, 240, 3451)-net over F4, using
- net defined by OOA [i] based on linear OOA(4240, 3451, F4, 39, 39) (dual of [(3451, 39), 134349, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4240, 65570, F4, 39) (dual of [65570, 65330, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4240, 65575, F4, 39) (dual of [65575, 65335, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(33) [i] based on
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4201, 65536, F4, 34) (dual of [65536, 65335, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(38) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(4240, 65575, F4, 39) (dual of [65575, 65335, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4240, 65570, F4, 39) (dual of [65570, 65330, 40]-code), using
- net defined by OOA [i] based on linear OOA(4240, 3451, F4, 39, 39) (dual of [(3451, 39), 134349, 40]-NRT-code), using
(243−39, 243, 42294)-Net over F4 — Digital
Digital (204, 243, 42294)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4243, 42294, F4, 39) (dual of [42294, 42051, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4243, 65586, F4, 39) (dual of [65586, 65343, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- linear OA(4233, 65536, F4, 39) (dual of [65536, 65303, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(4193, 65536, F4, 33) (dual of [65536, 65343, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(410, 50, F4, 5) (dual of [50, 40, 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(38) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(4243, 65586, F4, 39) (dual of [65586, 65343, 40]-code), using
(243−39, 243, large)-Net in Base 4 — Upper bound on s
There is no (204, 243, large)-net in base 4, because
- 37 times m-reduction [i] would yield (204, 206, large)-net in base 4, but