Best Known (69, 90, s)-Nets in Base 7
(69, 90, 1680)-Net over F7 — Constructive and digital
Digital (69, 90, 1680)-net over F7, using
- net defined by OOA [i] based on linear OOA(790, 1680, F7, 21, 21) (dual of [(1680, 21), 35190, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(790, 16801, F7, 21) (dual of [16801, 16711, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(790, 16806, F7, 21) (dual of [16806, 16716, 22]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(790, 16806, F7, 21) (dual of [16806, 16716, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(790, 16801, F7, 21) (dual of [16801, 16711, 22]-code), using
(69, 90, 12002)-Net over F7 — Digital
Digital (69, 90, 12002)-net over F7, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(790, 12002, F7, 21) (dual of [12002, 11912, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(790, 16806, F7, 21) (dual of [16806, 16716, 22]-code), using
- 1 times truncation [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]
- 1 times truncation [i] based on linear OA(791, 16807, F7, 22) (dual of [16807, 16716, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(790, 16806, F7, 21) (dual of [16806, 16716, 22]-code), using
(69, 90, large)-Net in Base 7 — Upper bound on s
There is no (69, 90, large)-net in base 7, because
- 19 times m-reduction [i] would yield (69, 71, large)-net in base 7, but