Best Known (126, 126+23, s)-Nets in Base 9
(126, 126+23, 434819)-Net over F9 — Constructive and digital
Digital (126, 149, 434819)-net over F9, using
- 91 times duplication [i] based on digital (125, 148, 434819)-net over F9, using
- net defined by OOA [i] based on linear OOA(9148, 434819, F9, 23, 23) (dual of [(434819, 23), 10000689, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9148, 4783010, F9, 23) (dual of [4783010, 4782862, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(9148, 4783011, F9, 23) (dual of [4783011, 4782863, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(9141, 4782969, F9, 23) (dual of [4782969, 4782828, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(9148, 4783011, F9, 23) (dual of [4783011, 4782863, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(9148, 4783010, F9, 23) (dual of [4783010, 4782862, 24]-code), using
- net defined by OOA [i] based on linear OOA(9148, 434819, F9, 23, 23) (dual of [(434819, 23), 10000689, 24]-NRT-code), using
(126, 126+23, 4783013)-Net over F9 — Digital
Digital (126, 149, 4783013)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(9149, 4783013, F9, 23) (dual of [4783013, 4782864, 24]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(9148, 4783011, F9, 23) (dual of [4783011, 4782863, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(9141, 4782969, F9, 23) (dual of [4782969, 4782828, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(9106, 4782969, F9, 17) (dual of [4782969, 4782863, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(97, 42, F9, 5) (dual of [42, 35, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [i] based on
- linear OA(9148, 4783012, F9, 22) (dual of [4783012, 4782864, 23]-code), using Gilbert–Varšamov bound and bm = 9148 > Vbs−1(k−1) = 33 904391 842963 516178 648037 196456 672429 215840 024387 254303 384994 167250 751203 983361 554694 686679 506729 708090 363049 291518 988258 360051 413542 683609 [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(9148, 4783011, F9, 23) (dual of [4783011, 4782863, 24]-code), using
- construction X with Varšamov bound [i] based on
(126, 126+23, large)-Net in Base 9 — Upper bound on s
There is no (126, 149, large)-net in base 9, because
- 21 times m-reduction [i] would yield (126, 128, large)-net in base 9, but