Best Known (102, 121, s)-Nets in Base 3
(102, 121, 6561)-Net over F3 — Constructive and digital
Digital (102, 121, 6561)-net over F3, using
- net defined by OOA [i] based on linear OOA(3121, 6561, F3, 19, 19) (dual of [(6561, 19), 124538, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 59050 | 320−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- OOA 9-folding and stacking with additional row [i] based on linear OA(3121, 59050, F3, 19) (dual of [59050, 58929, 20]-code), using
(102, 121, 18187)-Net over F3 — Digital
Digital (102, 121, 18187)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3121, 18187, F3, 3, 19) (dual of [(18187, 3), 54440, 20]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3121, 19683, F3, 3, 19) (dual of [(19683, 3), 58928, 20]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OOA 3-folding [i] based on linear OA(3121, 59049, F3, 19) (dual of [59049, 58928, 20]-code), using
- discarding factors / shortening the dual code based on linear OOA(3121, 19683, F3, 3, 19) (dual of [(19683, 3), 58928, 20]-NRT-code), using
(102, 121, 4768012)-Net in Base 3 — Upper bound on s
There is no (102, 121, 4768013)-net in base 3, because
- 1 times m-reduction [i] would yield (102, 120, 4768013)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1797 011157 593443 920202 801838 159455 292857 057112 033334 146043 > 3120 [i]