Best Known (124, 124+38, s)-Nets in Base 3
(124, 124+38, 464)-Net over F3 — Constructive and digital
Digital (124, 162, 464)-net over F3, using
- 32 times duplication [i] based on digital (122, 160, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 40, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- trace code for nets [i] based on digital (2, 40, 116)-net over F81, using
(124, 124+38, 918)-Net over F3 — Digital
Digital (124, 162, 918)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3162, 918, F3, 38) (dual of [918, 756, 39]-code), using
- 169 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0) [i] based on linear OA(3148, 735, F3, 38) (dual of [735, 587, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(3148, 729, F3, 38) (dual of [729, 581, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3142, 729, F3, 37) (dual of [729, 587, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- 169 step Varšamov–Edel lengthening with (ri) = (2, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 9 times 0, 1, 12 times 0, 1, 16 times 0, 1, 19 times 0, 1, 21 times 0, 1, 22 times 0, 1, 23 times 0, 1, 25 times 0) [i] based on linear OA(3148, 735, F3, 38) (dual of [735, 587, 39]-code), using
(124, 124+38, 46355)-Net in Base 3 — Upper bound on s
There is no (124, 162, 46356)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 196689 107766 838211 359468 460430 512798 291690 693387 537897 556859 902013 163304 599281 > 3162 [i]