Best Known (36−11, 36, s)-Nets in Base 64
(36−11, 36, 52494)-Net over F64 — Constructive and digital
Digital (25, 36, 52494)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 65)-net over F64, using
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 0 and N(F) ≥ 65, using
- the rational function field F64(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 64)-sequence over F64, using
- digital (20, 31, 52429)-net over F64, using
- net defined by OOA [i] based on linear OOA(6431, 52429, F64, 11, 11) (dual of [(52429, 11), 576688, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(6431, 262146, F64, 11) (dual of [262146, 262115, 12]-code), using
- net defined by OOA [i] based on linear OOA(6431, 52429, F64, 11, 11) (dual of [(52429, 11), 576688, 12]-NRT-code), using
- digital (0, 5, 65)-net over F64, using
(36−11, 36, 262212)-Net over F64 — Digital
Digital (25, 36, 262212)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6436, 262212, F64, 11) (dual of [262212, 262176, 12]-code), using
- (u, u+v)-construction [i] based on
- linear OA(645, 65, F64, 5) (dual of [65, 60, 6]-code or 65-arc in PG(4,64)), using
- extended Reed–Solomon code RSe(60,64) [i]
- the expurgated narrow-sense BCH-code C(I) with length 65 | 642−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- linear OA(645, 65, F64, 5) (dual of [65, 60, 6]-code or 65-arc in PG(4,64)), using
- (u, u+v)-construction [i] based on
(36−11, 36, large)-Net in Base 64 — Upper bound on s
There is no (25, 36, large)-net in base 64, because
- 9 times m-reduction [i] would yield (25, 27, large)-net in base 64, but