Best Known (43, 43+10, s)-Nets in Base 7
(43, 43+10, 23534)-Net over F7 — Constructive and digital
Digital (43, 53, 23534)-net over F7, using
- net defined by OOA [i] based on linear OOA(753, 23534, F7, 10, 10) (dual of [(23534, 10), 235287, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(753, 117670, F7, 10) (dual of [117670, 117617, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(753, 117671, F7, 10) (dual of [117671, 117618, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(731, 117649, F7, 6) (dual of [117649, 117618, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(74, 22, F7, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,7)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(753, 117671, F7, 10) (dual of [117671, 117618, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(753, 117670, F7, 10) (dual of [117670, 117617, 11]-code), using
(43, 43+10, 117671)-Net over F7 — Digital
Digital (43, 53, 117671)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(753, 117671, F7, 10) (dual of [117671, 117618, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
- linear OA(749, 117649, F7, 10) (dual of [117649, 117600, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(731, 117649, F7, 6) (dual of [117649, 117618, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 117648 = 76−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(74, 22, F7, 3) (dual of [22, 18, 4]-code or 22-cap in PG(3,7)), using
- construction X applied to Ce(9) ⊂ Ce(5) [i] based on
(43, 43+10, large)-Net in Base 7 — Upper bound on s
There is no (43, 53, large)-net in base 7, because
- 8 times m-reduction [i] would yield (43, 45, large)-net in base 7, but