Best Known (144−26, 144, s)-Nets in Base 8
(144−26, 144, 20168)-Net over F8 — Constructive and digital
Digital (118, 144, 20168)-net over F8, using
- 81 times duplication [i] based on digital (117, 143, 20168)-net over F8, using
- net defined by OOA [i] based on linear OOA(8143, 20168, F8, 26, 26) (dual of [(20168, 26), 524225, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8143, 262184, F8, 26) (dual of [262184, 262041, 27]-code), using
- 3 times code embedding in larger space [i] based on linear OA(8140, 262181, F8, 26) (dual of [262181, 262041, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8103, 262144, F8, 20) (dual of [262144, 262041, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(87, 37, F8, 5) (dual of [37, 30, 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
- 3 times code embedding in larger space [i] based on linear OA(8140, 262181, F8, 26) (dual of [262181, 262041, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(8143, 262184, F8, 26) (dual of [262184, 262041, 27]-code), using
- net defined by OOA [i] based on linear OOA(8143, 20168, F8, 26, 26) (dual of [(20168, 26), 524225, 27]-NRT-code), using
(144−26, 144, 262192)-Net over F8 — Digital
Digital (118, 144, 262192)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8144, 262192, F8, 26) (dual of [262192, 262048, 27]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8143, 262190, F8, 26) (dual of [262190, 262047, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(8133, 262144, F8, 26) (dual of [262144, 262011, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(897, 262144, F8, 19) (dual of [262144, 262047, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(810, 46, F8, 6) (dual of [46, 36, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- a “GraX†code from Grassl’s database [i]
- discarding factors / shortening the dual code based on linear OA(810, 74, F8, 6) (dual of [74, 64, 7]-code), using
- construction X applied to Ce(25) ⊂ Ce(18) [i] based on
- linear OA(8143, 262191, F8, 25) (dual of [262191, 262048, 26]-code), using Gilbert–Varšamov bound and bm = 8143 > Vbs−1(k−1) = 3 435443 037854 812636 641920 945272 293024 945524 349056 170519 059767 000372 444226 177916 617673 221148 183325 674578 005394 178780 546673 890822 [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(8143, 262190, F8, 26) (dual of [262190, 262047, 27]-code), using
- construction X with Varšamov bound [i] based on
(144−26, 144, large)-Net in Base 8 — Upper bound on s
There is no (118, 144, large)-net in base 8, because
- 24 times m-reduction [i] would yield (118, 120, large)-net in base 8, but