Best Known (125−28, 125, s)-Nets in Base 3
(125−28, 125, 464)-Net over F3 — Constructive and digital
Digital (97, 125, 464)-net over F3, using
- 31 times duplication [i] based on digital (96, 124, 464)-net over F3, using
- t-expansion [i] based on digital (95, 124, 464)-net over F3, using
- trace code for nets [i] based on digital (2, 31, 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, 31, 116)-net over F81, using
- t-expansion [i] based on digital (95, 124, 464)-net over F3, using
(125−28, 125, 901)-Net over F3 — Digital
Digital (97, 125, 901)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3125, 901, F3, 28) (dual of [901, 776, 29]-code), using
- 145 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 22 times 0, 1, 26 times 0, 1, 30 times 0) [i] based on linear OA(3111, 742, F3, 28) (dual of [742, 631, 29]-code), using
- construction XX applied to C1 = C([342,367]), C2 = C([340,365]), C3 = C1 + C2 = C([342,365]), and C∩ = C1 ∩ C2 = C([340,367]) [i] based on
- 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 = {342,343,…,367}, and designed minimum distance d ≥ |I|+1 = 27 [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(3109, 728, F3, 28) (dual of [728, 619, 29]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {340,341,…,367}, and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(397, 728, F3, 24) (dual of [728, 631, 25]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {342,343,…,365}, and designed minimum distance d ≥ |I|+1 = 25 [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(31, 7, F3, 1) (dual of [7, 6, 2]-code) (see above)
- construction XX applied to C1 = C([342,367]), C2 = C([340,365]), C3 = C1 + C2 = C([342,365]), and C∩ = C1 ∩ C2 = C([340,367]) [i] based on
- 145 step Varšamov–Edel lengthening with (ri) = (3, 1, 1, 0, 1, 0, 0, 0, 1, 4 times 0, 1, 7 times 0, 1, 9 times 0, 1, 13 times 0, 1, 18 times 0, 1, 22 times 0, 1, 26 times 0, 1, 30 times 0) [i] based on linear OA(3111, 742, F3, 28) (dual of [742, 631, 29]-code), using
(125−28, 125, 54996)-Net in Base 3 — Upper bound on s
There is no (97, 125, 54997)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 436766 908370 349446 049721 644479 314222 308401 003372 497923 860097 > 3125 [i]