Best Known (131, 131+25, s)-Nets in Base 8
(131, 131+25, 174766)-Net over F8 — Constructive and digital
Digital (131, 156, 174766)-net over F8, using
- 81 times duplication [i] based on digital (130, 155, 174766)-net over F8, using
- net defined by OOA [i] based on linear OOA(8155, 174766, F8, 25, 25) (dual of [(174766, 25), 4368995, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8155, 2097193, F8, 25) (dual of [2097193, 2097038, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8155, 2097194, F8, 25) (dual of [2097194, 2097039, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(18) [i] based on
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8113, 2097152, F8, 19) (dual of [2097152, 2097039, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(24) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8155, 2097194, F8, 25) (dual of [2097194, 2097039, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(8155, 2097193, F8, 25) (dual of [2097193, 2097038, 26]-code), using
- net defined by OOA [i] based on linear OOA(8155, 174766, F8, 25, 25) (dual of [(174766, 25), 4368995, 26]-NRT-code), using
(131, 131+25, 1642086)-Net over F8 — Digital
Digital (131, 156, 1642086)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8156, 1642086, F8, 25) (dual of [1642086, 1641930, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(8156, 2097168, F8, 25) (dual of [2097168, 2097012, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(8155, 2097153, F8, 25) (dual of [2097153, 2096998, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(8141, 2097153, F8, 23) (dual of [2097153, 2097012, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 2097153 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 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([0,12]) ⊂ C([0,11]) [i] based on
- discarding factors / shortening the dual code based on linear OA(8156, 2097168, F8, 25) (dual of [2097168, 2097012, 26]-code), using
(131, 131+25, large)-Net in Base 8 — Upper bound on s
There is no (131, 156, large)-net in base 8, because
- 23 times m-reduction [i] would yield (131, 133, large)-net in base 8, but