Best Known (73, 73+11, s)-Nets in Base 16
(73, 73+11, 4399935)-Net over F16 — Constructive and digital
Digital (73, 84, 4399935)-net over F16, using
- (u, u+v)-construction [i] based on
- digital (17, 22, 1044495)-net over F16, using
- net defined by OOA [i] based on linear OOA(1622, 1044495, F16, 6, 5) (dual of [(1044495, 6), 6266948, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(1622, 1044496, F16, 2, 5) (dual of [(1044496, 2), 2088970, 6]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(160, s, F16, 2, 0) with arbitrarily large s, using
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code) (see above)
- linear OOA(161, 65281, F16, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(161, s, F16, 2, 1) with arbitrarily large s, using
- appending 1 arbitrary column [i] based on linear OA(161, s, F16, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- discarding factors / shortening the dual code based on linear OOA(161, s, F16, 2, 1) with arbitrarily large s, using
- linear OOA(161, 65281, F16, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code) (see above)
- linear OOA(161, 65281, F16, 2, 1) (dual of [(65281, 2), 130561, 2]-NRT-code) (see above)
- linear OOA(165, 65281, F16, 2, 2) (dual of [(65281, 2), 130557, 3]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(165, 69905, F16, 2, 2) (dual of [(69905, 2), 139805, 3]-NRT-code), using
- appending kth column [i] based on linear OA(165, 69905, F16, 2) (dual of [69905, 69900, 3]-code), using
- Hamming code H(5,16) [i]
- appending kth column [i] based on linear OA(165, 69905, F16, 2) (dual of [69905, 69900, 3]-code), using
- discarding factors / shortening the dual code based on linear OOA(165, 69905, F16, 2, 2) (dual of [(69905, 2), 139805, 3]-NRT-code), using
- linear OOA(1614, 65281, F16, 2, 5) (dual of [(65281, 2), 130548, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- trace code [i] based on linear OA(2567, 65281, F256, 5) (dual of [65281, 65274, 6]-code), using
- OOA 2-folding [i] based on linear OA(1614, 130562, F16, 5) (dual of [130562, 130548, 6]-code), using
- linear OOA(160, 65281, F16, 2, 0) (dual of [(65281, 2), 130562, 1]-NRT-code), using
- generalized (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(1622, 1044496, F16, 2, 5) (dual of [(1044496, 2), 2088970, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(1622, 1044495, F16, 6, 5) (dual of [(1044495, 6), 6266948, 6]-NRT-code), using
- digital (51, 62, 3355440)-net over F16, using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 2563−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- discarding factors / shortening the dual code based on linear OA(25631, large, F256, 11) (dual of [large, large−31, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(25631, 8388601, F256, 11) (dual of [8388601, 8388570, 12]-code), using
- net defined by OOA [i] based on linear OOA(25631, 1677720, F256, 11, 11) (dual of [(1677720, 11), 18454889, 12]-NRT-code), using
- trace code for nets [i] based on digital (20, 31, 1677720)-net over F256, using
- digital (17, 22, 1044495)-net over F16, using
(73, 73+11, large)-Net over F16 — Digital
Digital (73, 84, large)-net over F16, using
- t-expansion [i] based on digital (70, 84, large)-net over F16, using
- 1 times m-reduction [i] based on digital (70, 85, large)-net over F16, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1685, large, F16, 15) (dual of [large, large−85, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 1612−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(1685, large, F16, 15) (dual of [large, large−85, 16]-code), using
- 1 times m-reduction [i] based on digital (70, 85, large)-net over F16, using
(73, 73+11, large)-Net in Base 16 — Upper bound on s
There is no (73, 84, large)-net in base 16, because
- 9 times m-reduction [i] would yield (73, 75, large)-net in base 16, but