Best Known (123−22, 123, s)-Nets in Base 8
(123−22, 123, 23834)-Net over F8 — Constructive and digital
Digital (101, 123, 23834)-net over F8, using
- 83 times duplication [i] based on digital (98, 120, 23834)-net over F8, using
- net defined by OOA [i] based on linear OOA(8120, 23834, F8, 22, 22) (dual of [(23834, 22), 524228, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8120, 262174, F8, 22) (dual of [262174, 262054, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- linear OA(8115, 262144, F8, 22) (dual of [262144, 262029, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(891, 262144, F8, 18) (dual of [262144, 262053, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(885, 262144, F8, 17) (dual of [262144, 262059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(84, 29, F8, 3) (dual of [29, 25, 4]-code or 29-cap in PG(3,8)), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(21) ⊂ Ce(17) ⊂ Ce(16) [i] based on
- OA 11-folding and stacking [i] based on linear OA(8120, 262174, F8, 22) (dual of [262174, 262054, 23]-code), using
- net defined by OOA [i] based on linear OOA(8120, 23834, F8, 22, 22) (dual of [(23834, 22), 524228, 23]-NRT-code), using
(123−22, 123, 262184)-Net over F8 — Digital
Digital (101, 123, 262184)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8123, 262184, F8, 22) (dual of [262184, 262061, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8121, 262180, F8, 22) (dual of [262180, 262059, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(8115, 262144, F8, 22) (dual of [262144, 262029, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(885, 262144, F8, 17) (dual of [262144, 262059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(86, 36, F8, 4) (dual of [36, 30, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- 1 times truncation [i] based on linear OA(87, 57, F8, 5) (dual of [57, 50, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(86, 56, F8, 4) (dual of [56, 50, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(8121, 262182, F8, 21) (dual of [262182, 262061, 22]-code), using Gilbert–Varšamov bound and bm = 8121 > Vbs−1(k−1) = 77188 125050 973760 243681 891767 763044 522916 608971 787334 412893 106566 066404 108356 229360 688085 547352 251764 256254 [i]
- linear OA(80, 2, F8, 0) (dual of [2, 2, 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(8121, 262180, F8, 22) (dual of [262180, 262059, 23]-code), using
- construction X with Varšamov bound [i] based on
(123−22, 123, large)-Net in Base 8 — Upper bound on s
There is no (101, 123, large)-net in base 8, because
- 20 times m-reduction [i] would yield (101, 103, large)-net in base 8, but