Best Known (127, 127+26, s)-Nets in Base 8
(127, 127+26, 40330)-Net over F8 — Constructive and digital
Digital (127, 153, 40330)-net over F8, using
- 81 times duplication [i] based on digital (126, 152, 40330)-net over F8, using
- net defined by OOA [i] based on linear OOA(8152, 40330, F8, 26, 26) (dual of [(40330, 26), 1048428, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8152, 524290, F8, 26) (dual of [524290, 524138, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8152, 524294, F8, 26) (dual of [524294, 524142, 27]-code), using
- trace code [i] based on linear OA(6476, 262147, F64, 26) (dual of [262147, 262071, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(6476, 262144, F64, 26) (dual of [262144, 262068, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- trace code [i] based on linear OA(6476, 262147, F64, 26) (dual of [262147, 262071, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8152, 524294, F8, 26) (dual of [524294, 524142, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8152, 524290, F8, 26) (dual of [524290, 524138, 27]-code), using
- net defined by OOA [i] based on linear OOA(8152, 40330, F8, 26, 26) (dual of [(40330, 26), 1048428, 27]-NRT-code), using
(127, 127+26, 524296)-Net over F8 — Digital
Digital (127, 153, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8153, 524296, F8, 26) (dual of [524296, 524143, 27]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8152, 524294, F8, 26) (dual of [524294, 524142, 27]-code), using
- trace code [i] based on linear OA(6476, 262147, F64, 26) (dual of [262147, 262071, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- linear OA(6476, 262144, F64, 26) (dual of [262144, 262068, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(6473, 262144, F64, 25) (dual of [262144, 262071, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(25) ⊂ Ce(24) [i] based on
- trace code [i] based on linear OA(6476, 262147, F64, 26) (dual of [262147, 262071, 27]-code), using
- linear OA(8152, 524295, F8, 25) (dual of [524295, 524143, 26]-code), using Gilbert–Varšamov bound and bm = 8152 > Vbs−1(k−1) = 57 440579 048489 590395 728591 559177 309573 817924 184674 721800 974154 144732 638453 900921 228904 690650 943998 461124 022978 462582 474045 732765 499392 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 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(8152, 524294, F8, 26) (dual of [524294, 524142, 27]-code), using
- construction X with Varšamov bound [i] based on
(127, 127+26, large)-Net in Base 8 — Upper bound on s
There is no (127, 153, large)-net in base 8, because
- 24 times m-reduction [i] would yield (127, 129, large)-net in base 8, but