Best Known (137, 137+26, s)-Nets in Base 8
(137, 137+26, 161322)-Net over F8 — Constructive and digital
Digital (137, 163, 161322)-net over F8, using
- 82 times duplication [i] based on digital (135, 161, 161322)-net over F8, using
- net defined by OOA [i] based on linear OOA(8161, 161322, F8, 26, 26) (dual of [(161322, 26), 4194211, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8161, 2097186, F8, 26) (dual of [2097186, 2097025, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- linear OA(8155, 2097152, F8, 26) (dual of [2097152, 2096997, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8127, 2097152, F8, 21) (dual of [2097152, 2097025, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(86, 34, F8, 4) (dual of [34, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- OA 13-folding and stacking [i] based on linear OA(8161, 2097186, F8, 26) (dual of [2097186, 2097025, 27]-code), using
- net defined by OOA [i] based on linear OOA(8161, 161322, F8, 26, 26) (dual of [(161322, 26), 4194211, 27]-NRT-code), using
(137, 137+26, 1746299)-Net over F8 — Digital
Digital (137, 163, 1746299)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8163, 1746299, F8, 26) (dual of [1746299, 1746136, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 2097195, F8, 26) (dual of [2097195, 2097032, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8162, 2097194, F8, 26) (dual of [2097194, 2097032, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(8155, 2097152, F8, 26) (dual of [2097152, 2096997, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8120, 2097152, F8, 20) (dual of [2097152, 2097032, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(87, 42, F8, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(8162, 2097194, F8, 26) (dual of [2097194, 2097032, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 2097195, F8, 26) (dual of [2097195, 2097032, 27]-code), using
(137, 137+26, large)-Net in Base 8 — Upper bound on s
There is no (137, 163, large)-net in base 8, because
- 24 times m-reduction [i] would yield (137, 139, large)-net in base 8, but