Best Known (67−18, 67, s)-Nets in Base 3
(67−18, 67, 192)-Net over F3 — Constructive and digital
Digital (49, 67, 192)-net over F3, using
- 31 times duplication [i] based on digital (48, 66, 192)-net over F3, using
- trace code for nets [i] based on digital (4, 22, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- trace code for nets [i] based on digital (4, 22, 64)-net over F27, using
(67−18, 67, 287)-Net over F3 — Digital
Digital (49, 67, 287)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(367, 287, F3, 18) (dual of [287, 220, 19]-code), using
- 29 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0) [i] based on linear OA(361, 252, F3, 18) (dual of [252, 191, 19]-code), using
- construction XX applied to C1 = C([241,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([241,16]) [i] based on
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,15}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(356, 242, F3, 17) (dual of [242, 186, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(361, 242, F3, 18) (dual of [242, 181, 19]-code), using the primitive BCH-code C(I) with length 242 = 35−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(351, 242, F3, 16) (dual of [242, 191, 17]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [0,15], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(30, 5, F3, 0) (dual of [5, 5, 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, 5, F3, 0) (dual of [5, 5, 1]-code) (see above)
- construction XX applied to C1 = C([241,15]), C2 = C([0,16]), C3 = C1 + C2 = C([0,15]), and C∩ = C1 ∩ C2 = C([241,16]) [i] based on
- 29 step Varšamov–Edel lengthening with (ri) = (2, 0, 1, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 10 times 0) [i] based on linear OA(361, 252, F3, 18) (dual of [252, 191, 19]-code), using
(67−18, 67, 7381)-Net in Base 3 — Upper bound on s
There is no (49, 67, 7382)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 92 804524 648572 284169 368929 380125 > 367 [i]