Best Known (121−21, 121, s)-Nets in Base 3
(121−21, 121, 688)-Net over F3 — Constructive and digital
Digital (100, 121, 688)-net over F3, using
- 3 times m-reduction [i] based on digital (100, 124, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 31, 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
- trace code for nets [i] based on digital (7, 31, 172)-net over F81, using
(121−21, 121, 4071)-Net over F3 — Digital
Digital (100, 121, 4071)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3121, 4071, F3, 21) (dual of [4071, 3950, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3121, 6590, F3, 21) (dual of [6590, 6469, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(3113, 6562, F3, 21) (dual of [6562, 6449, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(381, 6562, F3, 15) (dual of [6562, 6481, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 316−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3121, 6590, F3, 21) (dual of [6590, 6469, 22]-code), using
(121−21, 121, 1203366)-Net in Base 3 — Upper bound on s
There is no (100, 121, 1203367)-net in base 3, because
- 1 times m-reduction [i] would yield (100, 120, 1203367)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1797 016998 869510 402461 061732 294052 217990 276022 930787 076797 > 3120 [i]