Best Known (20, 28, s)-Nets in Base 7
(20, 28, 603)-Net over F7 — Constructive and digital
Digital (20, 28, 603)-net over F7, using
- net defined by OOA [i] based on linear OOA(728, 603, F7, 8, 8) (dual of [(603, 8), 4796, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(728, 2412, F7, 8) (dual of [2412, 2384, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(717, 2401, F7, 5) (dual of [2401, 2384, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(73, 11, F7, 2) (dual of [11, 8, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 48 = 72−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(73, 48, F7, 2) (dual of [48, 45, 3]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- OA 4-folding and stacking [i] based on linear OA(728, 2412, F7, 8) (dual of [2412, 2384, 9]-code), using
(20, 28, 2441)-Net over F7 — Digital
Digital (20, 28, 2441)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(728, 2441, F7, 8) (dual of [2441, 2413, 9]-code), using
- 37 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 30 times 0) [i] based on linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- 37 step Varšamov–Edel lengthening with (ri) = (2, 5 times 0, 1, 30 times 0) [i] based on linear OA(725, 2401, F7, 8) (dual of [2401, 2376, 9]-code), using
(20, 28, 303797)-Net in Base 7 — Upper bound on s
There is no (20, 28, 303798)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 459987 246736 278522 844969 > 728 [i]