Best Known (52, 52+22, s)-Nets in Base 7
(52, 52+22, 218)-Net over F7 — Constructive and digital
Digital (52, 74, 218)-net over F7, using
- 71 times duplication [i] based on digital (51, 73, 218)-net over F7, using
- net defined by OOA [i] based on linear OOA(773, 218, F7, 22, 22) (dual of [(218, 22), 4723, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(773, 2398, F7, 22) (dual of [2398, 2325, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- discarding factors / shortening the dual code based on linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(773, 2398, F7, 22) (dual of [2398, 2325, 23]-code), using
- net defined by OOA [i] based on linear OOA(773, 218, F7, 22, 22) (dual of [(218, 22), 4723, 23]-NRT-code), using
(52, 52+22, 1670)-Net over F7 — Digital
Digital (52, 74, 1670)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(774, 1670, F7, 22) (dual of [1670, 1596, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(774, 2406, F7, 22) (dual of [2406, 2332, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(19) [i] based on
- linear OA(773, 2401, F7, 22) (dual of [2401, 2328, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(769, 2401, F7, 20) (dual of [2401, 2332, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(71, 5, F7, 1) (dual of [5, 4, 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(21) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(774, 2406, F7, 22) (dual of [2406, 2332, 23]-code), using
(52, 52+22, 396333)-Net in Base 7 — Upper bound on s
There is no (52, 74, 396334)-net in base 7, because
- the generalized Rao bound for nets shows that 7m ≥ 344 560140 355364 688996 429462 360835 576986 632974 958068 126985 830585 > 774 [i]