Best Known (246−39, 246, s)-Nets in Base 4
(246−39, 246, 3452)-Net over F4 — Constructive and digital
Digital (207, 246, 3452)-net over F4, using
- net defined by OOA [i] based on linear OOA(4246, 3452, F4, 39, 39) (dual of [(3452, 39), 134382, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4246, 65589, F4, 39) (dual of [65589, 65343, 40]-code), using
- 3 times code embedding in larger space [i] 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
- 3 times code embedding in larger space [i] based on linear OA(4243, 65586, F4, 39) (dual of [65586, 65343, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(4246, 65589, F4, 39) (dual of [65589, 65343, 40]-code), using
(246−39, 246, 47329)-Net over F4 — Digital
Digital (207, 246, 47329)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(4246, 47329, F4, 39) (dual of [47329, 47083, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(4246, 65574, F4, 39) (dual of [65574, 65328, 40]-code), using
- construction X applied to C([0,20]) ⊂ C([0,17]) [i] based on
- linear OA(4241, 65537, F4, 41) (dual of [65537, 65296, 42]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,20], and minimum distance d ≥ |{−20,−19,…,20}|+1 = 42 (BCH-bound) [i]
- linear OA(4209, 65537, F4, 35) (dual of [65537, 65328, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 416−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(45, 37, F4, 3) (dual of [37, 32, 4]-code or 37-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 C([0,20]) ⊂ C([0,17]) [i] based on
- discarding factors / shortening the dual code based on linear OA(4246, 65574, F4, 39) (dual of [65574, 65328, 40]-code), using
(246−39, 246, large)-Net in Base 4 — Upper bound on s
There is no (207, 246, large)-net in base 4, because
- 37 times m-reduction [i] would yield (207, 209, large)-net in base 4, but