Best Known (101, 101+23, s)-Nets in Base 9
(101, 101+23, 48314)-Net over F9 — Constructive and digital
Digital (101, 124, 48314)-net over F9, using
- 92 times duplication [i] based on digital (99, 122, 48314)-net over F9, using
- net defined by OOA [i] based on linear OOA(9122, 48314, F9, 23, 23) (dual of [(48314, 23), 1111100, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9122, 531455, F9, 23) (dual of [531455, 531333, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,10]) [i] based on
- linear OA(9121, 531442, F9, 23) (dual of [531442, 531321, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(9109, 531442, F9, 21) (dual of [531442, 531333, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 531442 | 912−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(91, 13, F9, 1) (dual of [13, 12, 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([0,11]) ⊂ C([0,10]) [i] based on
- OOA 11-folding and stacking with additional row [i] based on linear OA(9122, 531455, F9, 23) (dual of [531455, 531333, 24]-code), using
- net defined by OOA [i] based on linear OOA(9122, 48314, F9, 23, 23) (dual of [(48314, 23), 1111100, 24]-NRT-code), using
(101, 101+23, 421235)-Net over F9 — Digital
Digital (101, 124, 421235)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9124, 421235, F9, 23) (dual of [421235, 421111, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9124, 531462, F9, 23) (dual of [531462, 531338, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(9121, 531441, F9, 23) (dual of [531441, 531320, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(9103, 531441, F9, 20) (dual of [531441, 531338, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 531440 = 96−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(93, 21, F9, 2) (dual of [21, 18, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 92−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(93, 80, F9, 2) (dual of [80, 77, 3]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(9124, 531462, F9, 23) (dual of [531462, 531338, 24]-code), using
(101, 101+23, large)-Net in Base 9 — Upper bound on s
There is no (101, 124, large)-net in base 9, because
- 21 times m-reduction [i] would yield (101, 103, large)-net in base 9, but