Best Known (124−32, 124, s)-Nets in Base 3
(124−32, 124, 264)-Net over F3 — Constructive and digital
Digital (92, 124, 264)-net over F3, using
- 31 times duplication [i] based on digital (91, 123, 264)-net over F3, using
- trace code for nets [i] based on digital (9, 41, 88)-net over F27, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 9 and N(F) ≥ 88, using
- net from sequence [i] based on digital (9, 87)-sequence over F27, using
- trace code for nets [i] based on digital (9, 41, 88)-net over F27, using
(124−32, 124, 519)-Net over F3 — Digital
Digital (92, 124, 519)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3124, 519, F3, 32) (dual of [519, 395, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- discarding factors / shortening the dual code based on linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using
(124−32, 124, 16935)-Net in Base 3 — Upper bound on s
There is no (92, 124, 16936)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 145608 942712 750153 402764 982977 920965 223533 060732 560771 894273 > 3124 [i]