Best Known (49, 49+11, s)-Nets in Base 8
(49, 49+11, 52438)-Net over F8 — Constructive and digital
Digital (49, 60, 52438)-net over F8, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 9)-net over F8, using
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F8 with g(F) = 0 and N(F) ≥ 9, using
- the rational function field F8(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 8)-sequence over F8, using
- digital (44, 55, 52429)-net over F8, using
- net defined by OOA [i] based on linear OOA(855, 52429, F8, 11, 11) (dual of [(52429, 11), 576664, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(855, 262146, F8, 11) (dual of [262146, 262091, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(855, 262150, F8, 11) (dual of [262150, 262095, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(855, 262144, F8, 11) (dual of [262144, 262089, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(849, 262144, F8, 10) (dual of [262144, 262095, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(80, 6, F8, 0) (dual of [6, 6, 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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(855, 262150, F8, 11) (dual of [262150, 262095, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(855, 262146, F8, 11) (dual of [262146, 262091, 12]-code), using
- net defined by OOA [i] based on linear OOA(855, 52429, F8, 11, 11) (dual of [(52429, 11), 576664, 12]-NRT-code), using
- digital (0, 5, 9)-net over F8, using
(49, 49+11, 262168)-Net over F8 — Digital
Digital (49, 60, 262168)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(860, 262168, F8, 11) (dual of [262168, 262108, 12]-code), using
- construction XX applied to Ce(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
- linear OA(855, 262144, F8, 11) (dual of [262144, 262089, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(837, 262144, F8, 7) (dual of [262144, 262107, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(831, 262144, F8, 6) (dual of [262144, 262113, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 86−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(84, 23, F8, 3) (dual of [23, 19, 4]-code or 23-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(10) ⊂ Ce(6) ⊂ Ce(5) [i] based on
(49, 49+11, large)-Net in Base 8 — Upper bound on s
There is no (49, 60, large)-net in base 8, because
- 9 times m-reduction [i] would yield (49, 51, large)-net in base 8, but