Best Known (139−34, 139, s)-Nets in Base 3
(139−34, 139, 328)-Net over F3 — Constructive and digital
Digital (105, 139, 328)-net over F3, using
- 1 times m-reduction [i] based on digital (105, 140, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 35, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 35, 82)-net over F81, using
(139−34, 139, 702)-Net over F3 — Digital
Digital (105, 139, 702)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3139, 702, F3, 34) (dual of [702, 563, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3139, 763, F3, 34) (dual of [763, 624, 35]-code), using
- construction XX applied to C1 = C([334,365]), C2 = C([340,367]), C3 = C1 + C2 = C([340,365]), and C∩ = C1 ∩ C2 = C([334,367]) [i] based on
- linear OA(3124, 728, F3, 32) (dual of [728, 604, 33]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,365}, and designed minimum distance d ≥ |I|+1 = 33 [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(3130, 728, F3, 34) (dual of [728, 598, 35]-code), using the primitive BCH-code C(I) with length 728 = 36−1, defining interval I = {334,335,…,367}, and designed minimum distance d ≥ |I|+1 = 35 [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(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- 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
- construction XX applied to C1 = C([334,365]), C2 = C([340,367]), C3 = C1 + C2 = C([340,365]), and C∩ = C1 ∩ C2 = C([334,367]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3139, 763, F3, 34) (dual of [763, 624, 35]-code), using
(139−34, 139, 28564)-Net in Base 3 — Upper bound on s
There is no (105, 139, 28565)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2 088746 226373 530310 205852 704497 663115 422886 489253 217355 374425 271531 > 3139 [i]