Best Known (157−26, 157, s)-Nets in Base 8
(157−26, 157, 161320)-Net over F8 — Constructive and digital
Digital (131, 157, 161320)-net over F8, using
- 81 times duplication [i] based on digital (130, 156, 161320)-net over F8, using
- net defined by OOA [i] based on linear OOA(8156, 161320, F8, 26, 26) (dual of [(161320, 26), 4194164, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8156, 2097160, F8, 26) (dual of [2097160, 2097004, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(8155, 2097159, F8, 26) (dual of [2097159, 2097004, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [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(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(80, 7, F8, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(8155, 2097159, F8, 26) (dual of [2097159, 2097004, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8156, 2097160, F8, 26) (dual of [2097160, 2097004, 27]-code), using
- net defined by OOA [i] based on linear OOA(8156, 161320, F8, 26, 26) (dual of [(161320, 26), 4194164, 27]-NRT-code), using
(157−26, 157, 1048581)-Net over F8 — Digital
Digital (131, 157, 1048581)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8157, 1048581, F8, 2, 26) (dual of [(1048581, 2), 2097005, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8157, 2097162, F8, 26) (dual of [2097162, 2097005, 27]-code), using
- construction XX applied to Ce(25) ⊂ Ce(24) ⊂ Ce(22) [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(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(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(80, 8, F8, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(81, 2, F8, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(25) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- OOA 2-folding [i] based on linear OA(8157, 2097162, F8, 26) (dual of [2097162, 2097005, 27]-code), using
(157−26, 157, large)-Net in Base 8 — Upper bound on s
There is no (131, 157, large)-net in base 8, because
- 24 times m-reduction [i] would yield (131, 133, large)-net in base 8, but