Best Known (80, 80+23, s)-Nets in Base 3
(80, 80+23, 464)-Net over F3 — Constructive and digital
Digital (80, 103, 464)-net over F3, using
- 1 times m-reduction [i] based on digital (80, 104, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 26, 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, 26, 116)-net over F81, using
(80, 80+23, 821)-Net over F3 — Digital
Digital (80, 103, 821)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3103, 821, F3, 23) (dual of [821, 718, 24]-code), using
- 69 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 14 times 0, 1, 20 times 0) [i] based on linear OA(392, 741, F3, 23) (dual of [741, 649, 24]-code), using
- construction XX applied to C1 = C([343,364]), C2 = C([345,365]), C3 = C1 + C2 = C([345,364]), and C∩ = C1 ∩ C2 = C([343,365]) [i] based on
- linear OA(385, 728, F3, 22) (dual of [728, 643, 23]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {343,344,…,364}, and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(385, 728, F3, 21) (dual of [728, 643, 22]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {345,346,…,365}, and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(391, 728, F3, 23) (dual of [728, 637, 24]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {343,344,…,365}, and designed minimum distance d ≥ |I|+1 = 24 [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 = {345,346,…,364}, and designed minimum distance d ≥ |I|+1 = 21 [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([343,364]), C2 = C([345,365]), C3 = C1 + C2 = C([345,364]), and C∩ = C1 ∩ C2 = C([343,365]) [i] based on
- 69 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 11 times 0, 1, 14 times 0, 1, 20 times 0) [i] based on linear OA(392, 741, F3, 23) (dual of [741, 649, 24]-code), using
(80, 80+23, 65182)-Net in Base 3 — Upper bound on s
There is no (80, 103, 65183)-net in base 3, because
- 1 times m-reduction [i] would yield (80, 102, 65183)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 4 638968 815539 106798 286489 684643 555600 163577 827947 > 3102 [i]