Best Known (113−25, 113, s)-Nets in Base 9
(113−25, 113, 4921)-Net over F9 — Constructive and digital
Digital (88, 113, 4921)-net over F9, using
- 92 times duplication [i] based on digital (86, 111, 4921)-net over F9, using
- net defined by OOA [i] based on linear OOA(9111, 4921, F9, 25, 25) (dual of [(4921, 25), 122914, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9111, 59053, F9, 25) (dual of [59053, 58942, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9111, 59054, F9, 25) (dual of [59054, 58943, 26]-code), using
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- linear OA(9111, 59049, F9, 25) (dual of [59049, 58938, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(9106, 59049, F9, 24) (dual of [59049, 58943, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(90, 5, F9, 0) (dual of [5, 5, 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
- construction X applied to Ce(24) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(9111, 59054, F9, 25) (dual of [59054, 58943, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(9111, 59053, F9, 25) (dual of [59053, 58942, 26]-code), using
- net defined by OOA [i] based on linear OOA(9111, 4921, F9, 25, 25) (dual of [(4921, 25), 122914, 26]-NRT-code), using
(113−25, 113, 52240)-Net over F9 — Digital
Digital (88, 113, 52240)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9113, 52240, F9, 25) (dual of [52240, 52127, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9113, 59062, F9, 25) (dual of [59062, 58949, 26]-code), using
- 1 times code embedding in larger space [i] based on linear OA(9112, 59061, F9, 25) (dual of [59061, 58949, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(9111, 59050, F9, 25) (dual of [59050, 58939, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(9101, 59050, F9, 23) (dual of [59050, 58949, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(91, 11, F9, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(91, s, F9, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(9112, 59061, F9, 25) (dual of [59061, 58949, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(9113, 59062, F9, 25) (dual of [59062, 58949, 26]-code), using
(113−25, 113, large)-Net in Base 9 — Upper bound on s
There is no (88, 113, large)-net in base 9, because
- 23 times m-reduction [i] would yield (88, 90, large)-net in base 9, but