Best Known (125−30, 125, s)-Nets in Base 3
(125−30, 125, 400)-Net over F3 — Constructive and digital
Digital (95, 125, 400)-net over F3, using
- 31 times duplication [i] based on digital (94, 124, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 31, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 31, 100)-net over F81, using
(125−30, 125, 708)-Net over F3 — Digital
Digital (95, 125, 708)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3125, 708, F3, 30) (dual of [708, 583, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 757, F3, 30) (dual of [757, 632, 31]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3124, 756, F3, 30) (dual of [756, 632, 31]-code), using
- construction XX applied to C1 = C([336,363]), C2 = C([340,365]), C3 = C1 + C2 = C([340,363]), and C∩ = C1 ∩ C2 = C([336,365]) [i] based on
- linear OA(3111, 728, F3, 28) (dual of [728, 617, 29]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {336,337,…,363}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3103, 728, F3, 26) (dual of [728, 625, 27]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {340,341,…,365}, and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(3118, 728, F3, 30) (dual of [728, 610, 31]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {336,337,…,365}, and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(396, 728, F3, 24) (dual of [728, 632, 25]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {340,341,…,363}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- linear OA(31, 8, F3, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction XX applied to C1 = C([336,363]), C2 = C([340,365]), C3 = C1 + C2 = C([340,363]), and C∩ = C1 ∩ C2 = C([336,365]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3124, 756, F3, 30) (dual of [756, 632, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3125, 757, F3, 30) (dual of [757, 632, 31]-code), using
(125−30, 125, 30376)-Net in Base 3 — Upper bound on s
There is no (95, 125, 30377)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 436748 816534 052013 104218 746669 176286 715875 388620 483727 055627 > 3125 [i]