Best Known (54, 77, s)-Nets in Base 7
(54, 77, 218)-Net over F7 — Constructive and digital
Digital (54, 77, 218)-net over F7, using
- net defined by OOA [i] based on linear OOA(777, 218, F7, 23, 23) (dual of [(218, 23), 4937, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(777, 2399, F7, 23) (dual of [2399, 2322, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(777, 2399, F7, 23) (dual of [2399, 2322, 24]-code), using
(54, 77, 1642)-Net over F7 — Digital
Digital (54, 77, 1642)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(777, 1642, F7, 23) (dual of [1642, 1565, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using
- an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2400 = 74−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- discarding factors / shortening the dual code based on linear OA(777, 2401, F7, 23) (dual of [2401, 2324, 24]-code), using
(54, 77, 564568)-Net in Base 7 — Upper bound on s
There is no (54, 77, 564569)-net in base 7, because
- 1 times m-reduction [i] would yield (54, 76, 564569)-net in base 7, but
- the generalized Rao bound for nets shows that 7m ≥ 16883 137696 684564 373567 352410 369355 273095 946482 213585 653327 576855 > 776 [i]