Best Known (82−18, 82, s)-Nets in Base 3
(82−18, 82, 464)-Net over F3 — Constructive and digital
Digital (64, 82, 464)-net over F3, using
- 32 times duplication [i] based on digital (62, 80, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 20, 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, 20, 116)-net over F81, using
(82−18, 82, 800)-Net over F3 — Digital
Digital (64, 82, 800)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(382, 800, F3, 18) (dual of [800, 718, 19]-code), using
- 51 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 12 times 0, 1, 17 times 0) [i] based on linear OA(373, 740, F3, 18) (dual of [740, 667, 19]-code), using
- construction XX applied to C1 = C([727,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([727,16]) [i] based on
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(367, 728, F3, 17) (dual of [728, 661, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(373, 728, F3, 18) (dual of [728, 655, 19]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(361, 728, F3, 16) (dual of [728, 667, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [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
- linear OA(30, 6, F3, 0) (dual of [6, 6, 1]-code) (see above)
- construction XX applied to C1 = C([727,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([727,16]) [i] based on
- 51 step Varšamov–Edel lengthening with (ri) = (3, 1, 0, 1, 0, 0, 1, 4 times 0, 1, 8 times 0, 1, 12 times 0, 1, 17 times 0) [i] based on linear OA(373, 740, F3, 18) (dual of [740, 667, 19]-code), using
(82−18, 82, 46104)-Net in Base 3 — Upper bound on s
There is no (64, 82, 46105)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1330 330808 752703 348082 020139 363614 456723 > 382 [i]