Best Known (72, 72+22, s)-Nets in Base 7
(72, 72+22, 1529)-Net over F7 — Constructive and digital
Digital (72, 94, 1529)-net over F7, using
- net defined by OOA [i] based on linear OOA(794, 1529, F7, 22, 22) (dual of [(1529, 22), 33544, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(794, 16819, F7, 22) (dual of [16819, 16725, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(794, 16820, F7, 22) (dual of [16820, 16726, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(781, 16807, F7, 19) (dual of [16807, 16726, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(73, 13, F7, 2) (dual of [13, 10, 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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(794, 16820, F7, 22) (dual of [16820, 16726, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(794, 16819, F7, 22) (dual of [16819, 16725, 23]-code), using
(72, 72+22, 11760)-Net over F7 — Digital
Digital (72, 94, 11760)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(794, 11760, F7, 22) (dual of [11760, 11666, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(794, 16820, F7, 22) (dual of [16820, 16726, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(18) [i] based on
- linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(781, 16807, F7, 19) (dual of [16807, 16726, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(73, 13, F7, 2) (dual of [13, 10, 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(21) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(794, 16820, F7, 22) (dual of [16820, 16726, 23]-code), using
(72, 72+22, large)-Net in Base 7 — Upper bound on s
There is no (72, 94, large)-net in base 7, because
- 20 times m-reduction [i] would yield (72, 74, large)-net in base 7, but