Best Known (54, 54+21, s)-Nets in Base 64
(54, 54+21, 26319)-Net over F64 — Constructive and digital
Digital (54, 75, 26319)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (3, 13, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- digital (41, 62, 26215)-net over F64, using
- net defined by OOA [i] based on linear OOA(6462, 26215, F64, 21, 21) (dual of [(26215, 21), 550453, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6462, 262151, F64, 21) (dual of [262151, 262089, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(6462, 262152, F64, 21) (dual of [262152, 262090, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- linear OA(6461, 262145, F64, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(6455, 262145, F64, 19) (dual of [262145, 262090, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145 | 646−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,10]) ⊂ C([0,9]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6462, 262152, F64, 21) (dual of [262152, 262090, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(6462, 262151, F64, 21) (dual of [262151, 262089, 22]-code), using
- net defined by OOA [i] based on linear OOA(6462, 26215, F64, 21, 21) (dual of [(26215, 21), 550453, 22]-NRT-code), using
- digital (3, 13, 104)-net over F64, using
(54, 54+21, 209716)-Net in Base 64 — Constructive
(54, 75, 209716)-net in base 64, using
- 641 times duplication [i] based on (53, 74, 209716)-net in base 64, using
- net defined by OOA [i] based on OOA(6474, 209716, S64, 21, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6474, 2097161, S64, 21), using
- discarding factors based on OA(6474, 2097163, S64, 21), using
- discarding parts of the base [i] based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- linear OA(12861, 2097152, F128, 21) (dual of [2097152, 2097091, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(12852, 2097152, F128, 18) (dual of [2097152, 2097100, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(1282, 11, F128, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(20) ⊂ Ce(17) [i] based on
- discarding parts of the base [i] based on linear OA(12863, 2097163, F128, 21) (dual of [2097163, 2097100, 22]-code), using
- discarding factors based on OA(6474, 2097163, S64, 21), using
- OOA 10-folding and stacking with additional row [i] based on OA(6474, 2097161, S64, 21), using
- net defined by OOA [i] based on OOA(6474, 209716, S64, 21, 21), using
(54, 54+21, 781891)-Net over F64 — Digital
Digital (54, 75, 781891)-net over F64, using
(54, 54+21, large)-Net in Base 64 — Upper bound on s
There is no (54, 75, large)-net in base 64, because
- 19 times m-reduction [i] would yield (54, 56, large)-net in base 64, but