Best Known (91, 91+21, s)-Nets in Base 3
(91, 91+21, 688)-Net over F3 — Constructive and digital
Digital (91, 112, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 28, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
(91, 91+21, 3094)-Net over F3 — Digital
Digital (91, 112, 3094)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3112, 3094, F3, 2, 21) (dual of [(3094, 2), 6076, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3112, 3280, F3, 2, 21) (dual of [(3280, 2), 6448, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3112, 6560, F3, 21) (dual of [6560, 6448, 22]-code), using
- 1 times truncation [i] based on linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using
- OOA 2-folding [i] based on linear OA(3112, 6560, F3, 21) (dual of [6560, 6448, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(3112, 3280, F3, 2, 21) (dual of [(3280, 2), 6448, 22]-NRT-code), using
(91, 91+21, 447695)-Net in Base 3 — Upper bound on s
There is no (91, 112, 447696)-net in base 3, because
- 1 times m-reduction [i] would yield (91, 111, 447696)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 91297 990691 339108 447976 130852 599791 262247 253010 107361 > 3111 [i]