Best Known (89−20, 89, s)-Nets in Base 3
(89−20, 89, 464)-Net over F3 — Constructive and digital
Digital (69, 89, 464)-net over F3, using
- 31 times duplication [i] based on digital (68, 88, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 22, 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, 22, 116)-net over F81, using
(89−20, 89, 783)-Net over F3 — Digital
Digital (69, 89, 783)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(389, 783, F3, 20) (dual of [783, 694, 21]-code), using
- 33 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0) [i] based on linear OA(380, 741, F3, 20) (dual of [741, 661, 21]-code), using
- construction XX applied to C1 = C([346,364]), C2 = C([348,365]), C3 = C1 + C2 = C([348,364]), and C∩ = C1 ∩ C2 = C([346,365]) [i] based on
- linear OA(373, 728, F3, 19) (dual of [728, 655, 20]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {346,347,…,364}, and designed minimum distance d ≥ |I|+1 = 20 [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 = {348,349,…,365}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(379, 728, F3, 20) (dual of [728, 649, 21]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {346,347,…,365}, and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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 = {348,349,…,364}, and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(31, 7, F3, 1) (dual of [7, 6, 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
- 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 XX applied to C1 = C([346,364]), C2 = C([348,365]), C3 = C1 + C2 = C([348,364]), and C∩ = C1 ∩ C2 = C([346,365]) [i] based on
- 33 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 5 times 0, 1, 7 times 0, 1, 10 times 0) [i] based on linear OA(380, 741, F3, 20) (dual of [741, 661, 21]-code), using
(89−20, 89, 39922)-Net in Base 3 — Upper bound on s
There is no (69, 89, 39923)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 909389 210703 471966 284058 491736 777656 130101 > 389 [i]