Best Known (140−39, 140, s)-Nets in Base 9
(140−39, 140, 780)-Net over F9 — Constructive and digital
Digital (101, 140, 780)-net over F9, using
- 2 times m-reduction [i] based on digital (101, 142, 780)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (8, 28, 40)-net over F9, using
- net from sequence [i] based on digital (8, 39)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 8 and N(F) ≥ 40, using
- net from sequence [i] based on digital (8, 39)-sequence over F9, using
- digital (73, 114, 740)-net over F9, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 57, 370)-net over F81, using
- digital (8, 28, 40)-net over F9, using
- (u, u+v)-construction [i] based on
(140−39, 140, 6575)-Net over F9 — Digital
Digital (101, 140, 6575)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9140, 6575, F9, 39) (dual of [6575, 6435, 40]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9138, 6571, F9, 39) (dual of [6571, 6433, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(9137, 6562, F9, 39) (dual of [6562, 6425, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(9129, 6562, F9, 37) (dual of [6562, 6433, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 98−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(91, 9, F9, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(9138, 6573, F9, 38) (dual of [6573, 6435, 39]-code), using Gilbert–Varšamov bound and bm = 9138 > Vbs−1(k−1) = 306631 601467 269246 415836 279248 049964 720922 697007 489411 016616 623514 637615 953597 079742 957091 948579 527501 769598 485038 669800 687397 630433 [i]
- linear OA(90, 2, F9, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(9138, 6571, F9, 39) (dual of [6571, 6433, 40]-code), using
- construction X with Varšamov bound [i] based on
(140−39, 140, large)-Net in Base 9 — Upper bound on s
There is no (101, 140, large)-net in base 9, because
- 37 times m-reduction [i] would yield (101, 103, large)-net in base 9, but