Best Known (124−27, 124, s)-Nets in Base 3
(124−27, 124, 600)-Net over F3 — Constructive and digital
Digital (97, 124, 600)-net over F3, using
- trace code for nets [i] based on digital (4, 31, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
(124−27, 124, 1008)-Net over F3 — Digital
Digital (97, 124, 1008)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3124, 1008, F3, 27) (dual of [1008, 884, 28]-code), using
- 264 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 28 times 0, 1, 31 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0) [i] based on linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using
- the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 264 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0, 1, 14 times 0, 1, 19 times 0, 1, 24 times 0, 1, 28 times 0, 1, 31 times 0, 1, 34 times 0, 1, 37 times 0, 1, 39 times 0) [i] based on linear OA(3108, 728, F3, 27) (dual of [728, 620, 28]-code), using
(124−27, 124, 92601)-Net in Base 3 — Upper bound on s
There is no (97, 124, 92602)-net in base 3, because
- 1 times m-reduction [i] would yield (97, 123, 92602)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 48521 744625 606484 735849 164976 070943 978528 790740 217975 550589 > 3123 [i]