Best Known (93−23, 93, s)-Nets in Base 9
(93−23, 93, 1194)-Net over F9 — Constructive and digital
Digital (70, 93, 1194)-net over F9, using
- net defined by OOA [i] based on linear OOA(993, 1194, F9, 23, 23) (dual of [(1194, 23), 27369, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(993, 13135, F9, 23) (dual of [13135, 13042, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(992, 13134, F9, 23) (dual of [13134, 13042, 24]-code), using
- trace code [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- trace code [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- 1 times code embedding in larger space [i] based on linear OA(992, 13134, F9, 23) (dual of [13134, 13042, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(993, 13135, F9, 23) (dual of [13135, 13042, 24]-code), using
(93−23, 93, 13136)-Net over F9 — Digital
Digital (70, 93, 13136)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(993, 13136, F9, 23) (dual of [13136, 13043, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(992, 13134, F9, 23) (dual of [13134, 13042, 24]-code), using
- trace code [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(8145, 6562, F81, 23) (dual of [6562, 6517, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(8141, 6562, F81, 21) (dual of [6562, 6521, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 6562 | 814−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- trace code [i] based on linear OA(8146, 6567, F81, 23) (dual of [6567, 6521, 24]-code), using
- linear OA(992, 13135, F9, 22) (dual of [13135, 13043, 23]-code), using Gilbert–Varšamov bound and bm = 992 > Vbs−1(k−1) = 54 453078 812243 103384 477486 137889 149135 990583 712112 993013 357559 294839 274025 392619 229489 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(992, 13134, F9, 23) (dual of [13134, 13042, 24]-code), using
- construction X with Varšamov bound [i] based on
(93−23, 93, large)-Net in Base 9 — Upper bound on s
There is no (70, 93, large)-net in base 9, because
- 21 times m-reduction [i] would yield (70, 72, large)-net in base 9, but