Best Known (81, 108, s)-Nets in Base 9
(81, 108, 1010)-Net over F9 — Constructive and digital
Digital (81, 108, 1010)-net over F9, using
- net defined by OOA [i] based on linear OOA(9108, 1010, F9, 27, 27) (dual of [(1010, 27), 27162, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9108, 13131, F9, 27) (dual of [13131, 13023, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9108, 13134, F9, 27) (dual of [13134, 13026, 28]-code), using
- trace code [i] based on linear OA(8154, 6567, F81, 27) (dual of [6567, 6513, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- trace code [i] based on linear OA(8154, 6567, F81, 27) (dual of [6567, 6513, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(9108, 13134, F9, 27) (dual of [13134, 13026, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(9108, 13131, F9, 27) (dual of [13131, 13023, 28]-code), using
(81, 108, 13134)-Net over F9 — Digital
Digital (81, 108, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9108, 13134, F9, 27) (dual of [13134, 13026, 28]-code), using
- trace code [i] based on linear OA(8154, 6567, F81, 27) (dual of [6567, 6513, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- linear OA(8153, 6562, F81, 27) (dual of [6562, 6509, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(8149, 6562, F81, 25) (dual of [6562, 6513, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,13]) ⊂ C([0,12]) [i] based on
- trace code [i] based on linear OA(8154, 6567, F81, 27) (dual of [6567, 6513, 28]-code), using
(81, 108, large)-Net in Base 9 — Upper bound on s
There is no (81, 108, large)-net in base 9, because
- 25 times m-reduction [i] would yield (81, 83, large)-net in base 9, but