Best Known (72, 72+21, s)-Nets in Base 16
(72, 72+21, 13131)-Net over F16 — Constructive and digital
Digital (72, 93, 13131)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 24)-net over F16, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 1 and N(F) ≥ 24, using
- net from sequence [i] based on digital (1, 23)-sequence over F16, using
- digital (61, 82, 13107)-net over F16, using
- net defined by OOA [i] based on linear OOA(1682, 13107, F16, 21, 21) (dual of [(13107, 21), 275165, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1682, 131071, F16, 21) (dual of [131071, 130989, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 131074, F16, 21) (dual of [131074, 130992, 22]-code), using
- trace code [i] based on linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- trace code [i] based on linear OA(25641, 65537, F256, 21) (dual of [65537, 65496, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(1682, 131074, F16, 21) (dual of [131074, 130992, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(1682, 131071, F16, 21) (dual of [131071, 130989, 22]-code), using
- net defined by OOA [i] based on linear OOA(1682, 13107, F16, 21, 21) (dual of [(13107, 21), 275165, 22]-NRT-code), using
- digital (1, 11, 24)-net over F16, using
(72, 72+21, 26215)-Net in Base 16 — Constructive
(72, 93, 26215)-net in base 16, using
- base change [i] based on 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
(72, 72+21, 219985)-Net over F16 — Digital
Digital (72, 93, 219985)-net over F16, using
(72, 72+21, large)-Net in Base 16 — Upper bound on s
There is no (72, 93, large)-net in base 16, because
- 19 times m-reduction [i] would yield (72, 74, large)-net in base 16, but