Best Known (142−32, 142, s)-Nets in Base 8
(142−32, 142, 2048)-Net over F8 — Constructive and digital
Digital (110, 142, 2048)-net over F8, using
- 81 times duplication [i] based on digital (109, 141, 2048)-net over F8, using
- t-expansion [i] based on digital (108, 141, 2048)-net over F8, using
- net defined by OOA [i] based on linear OOA(8141, 2048, F8, 33, 33) (dual of [(2048, 33), 67443, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(8141, 32769, F8, 33) (dual of [32769, 32628, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 32769 | 810−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(8141, 32769, F8, 33) (dual of [32769, 32628, 34]-code), using
- net defined by OOA [i] based on linear OOA(8141, 2048, F8, 33, 33) (dual of [(2048, 33), 67443, 34]-NRT-code), using
- t-expansion [i] based on digital (108, 141, 2048)-net over F8, using
(142−32, 142, 30196)-Net over F8 — Digital
Digital (110, 142, 30196)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8142, 30196, F8, 32) (dual of [30196, 30054, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(8142, 32779, F8, 32) (dual of [32779, 32637, 33]-code), using
- construction X applied to Ce(32) ⊂ Ce(29) [i] based on
- linear OA(8141, 32768, F8, 33) (dual of [32768, 32627, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(8131, 32768, F8, 30) (dual of [32768, 32637, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 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 Ce(32) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(8142, 32779, F8, 32) (dual of [32779, 32637, 33]-code), using
(142−32, 142, large)-Net in Base 8 — Upper bound on s
There is no (110, 142, large)-net in base 8, because
- 30 times m-reduction [i] would yield (110, 112, large)-net in base 8, but