Best Known (172−39, 172, s)-Nets in Base 8
(172−39, 172, 1725)-Net over F8 — Constructive and digital
Digital (133, 172, 1725)-net over F8, using
- net defined by OOA [i] based on linear OOA(8172, 1725, F8, 39, 39) (dual of [(1725, 39), 67103, 40]-NRT-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(8172, 32776, F8, 39) (dual of [32776, 32604, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(8172, 32780, F8, 39) (dual of [32780, 32608, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(8171, 32769, F8, 39) (dual of [32769, 32598, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(8161, 32769, F8, 37) (dual of [32769, 32608, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 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
- discarding factors / shortening the dual code based on linear OA(8172, 32780, F8, 39) (dual of [32780, 32608, 40]-code), using
- OOA 19-folding and stacking with additional row [i] based on linear OA(8172, 32776, F8, 39) (dual of [32776, 32604, 40]-code), using
(172−39, 172, 31206)-Net over F8 — Digital
Digital (133, 172, 31206)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8172, 31206, F8, 39) (dual of [31206, 31034, 40]-code), using
- discarding factors / shortening the dual code based on linear OA(8172, 32780, F8, 39) (dual of [32780, 32608, 40]-code), using
- construction X applied to C([0,19]) ⊂ C([0,18]) [i] based on
- linear OA(8171, 32769, F8, 39) (dual of [32769, 32598, 40]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,19], and minimum distance d ≥ |{−19,−18,…,19}|+1 = 40 (BCH-bound) [i]
- linear OA(8161, 32769, F8, 37) (dual of [32769, 32608, 38]-code), using the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,18], and minimum distance d ≥ |{−18,−17,…,18}|+1 = 38 (BCH-bound) [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 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
- discarding factors / shortening the dual code based on linear OA(8172, 32780, F8, 39) (dual of [32780, 32608, 40]-code), using
(172−39, 172, large)-Net in Base 8 — Upper bound on s
There is no (133, 172, large)-net in base 8, because
- 37 times m-reduction [i] would yield (133, 135, large)-net in base 8, but