Best Known (114−11, 114, s)-Nets in Base 9
(114−11, 114, 5747018)-Net over F9 — Constructive and digital
Digital (103, 114, 5747018)-net over F9, using
- (u, u+v)-construction [i] based on
- digital (27, 32, 2391578)-net over F9, using
- net defined by OOA [i] based on linear OOA(932, 2391578, F9, 6, 5) (dual of [(2391578, 6), 14349436, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(932, 2391579, F9, 2, 5) (dual of [(2391579, 2), 4783126, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(93, 91, F9, 2, 2) (dual of [(91, 2), 179, 3]-NRT-code), using
- appending kth column [i] based on linear OA(93, 91, F9, 2) (dual of [91, 88, 3]-code), using
- Hamming code H(3,9) [i]
- appending kth column [i] based on linear OA(93, 91, F9, 2) (dual of [91, 88, 3]-code), using
- linear OOA(929, 2391488, F9, 2, 5) (dual of [(2391488, 2), 4782947, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(929, 4782976, F9, 5) (dual of [4782976, 4782947, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(929, 4782969, F9, 5) (dual of [4782969, 4782940, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(922, 4782969, F9, 4) (dual of [4782969, 4782947, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4782968 = 97−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(90, 7, F9, 0) (dual of [7, 7, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding [i] based on linear OA(929, 4782976, F9, 5) (dual of [4782976, 4782947, 6]-code), using
- linear OOA(93, 91, F9, 2, 2) (dual of [(91, 2), 179, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(932, 2391579, F9, 2, 5) (dual of [(2391579, 2), 4783126, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(932, 2391578, F9, 6, 5) (dual of [(2391578, 6), 14349436, 6]-NRT-code), using
- digital (71, 82, 3355440)-net over F9, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- digital (27, 32, 2391578)-net over F9, using
(114−11, 114, 6710880)-Net in Base 9 — Constructive
(103, 114, 6710880)-net in base 9, using
- (u, u+v)-construction [i] based on
- (27, 32, 4194301)-net in base 9, using
- net defined by OOA [i] based on OOA(932, 4194301, S9, 5, 5), using
- OOA 2-folding and stacking with additional row [i] based on OA(932, large, S9, 5), using
- discarding parts of the base [i] based on linear OA(2721, large, F27, 5) (dual of [large, large−21, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(2721, large, F27, 5) (dual of [large, large−21, 6]-code), using
- OOA 2-folding and stacking with additional row [i] based on OA(932, large, S9, 5), using
- net defined by OOA [i] based on OOA(932, 4194301, S9, 5, 5), using
- digital (71, 82, 3355440)-net over F9, using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(8141, large, F81, 11) (dual of [large, large−41, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(8141, 8388601, F81, 11) (dual of [8388601, 8388560, 12]-code), using
- net defined by OOA [i] based on linear OOA(8141, 1677720, F81, 11, 11) (dual of [(1677720, 11), 18454879, 12]-NRT-code), using
- trace code for nets [i] based on digital (30, 41, 1677720)-net over F81, using
- (27, 32, 4194301)-net in base 9, using
(114−11, 114, large)-Net over F9 — Digital
Digital (103, 114, large)-net over F9, using
- t-expansion [i] based on digital (101, 114, large)-net over F9, using
- 4 times m-reduction [i] based on digital (101, 118, large)-net over F9, using
(114−11, 114, large)-Net in Base 9 — Upper bound on s
There is no (103, 114, large)-net in base 9, because
- 9 times m-reduction [i] would yield (103, 105, large)-net in base 9, but