Best Known (32−8, 32, s)-Nets in Base 7
(32−8, 32, 4203)-Net over F7 — Constructive and digital
Digital (24, 32, 4203)-net over F7, using
- net defined by OOA [i] based on linear OOA(732, 4203, F7, 8, 8) (dual of [(4203, 8), 33592, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(732, 16812, F7, 8) (dual of [16812, 16780, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(732, 16813, F7, 8) (dual of [16813, 16781, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(726, 16807, F7, 6) (dual of [16807, 16781, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 6, F7, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(732, 16813, F7, 8) (dual of [16813, 16781, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(732, 16812, F7, 8) (dual of [16812, 16780, 9]-code), using
(32−8, 32, 11596)-Net over F7 — Digital
Digital (24, 32, 11596)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(732, 11596, F7, 8) (dual of [11596, 11564, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(732, 16813, F7, 8) (dual of [16813, 16781, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(731, 16807, F7, 8) (dual of [16807, 16776, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(726, 16807, F7, 6) (dual of [16807, 16781, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16806 = 75−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(71, 6, F7, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(71, s, F7, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(732, 16813, F7, 8) (dual of [16813, 16781, 9]-code), using
(32−8, 32, 2126598)-Net in Base 7 — Upper bound on s
There is no (24, 32, 2126599)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 1104 429379 497813 276965 702377 > 732 [i]