Best Known (19, 19+14, s)-Nets in Base 128
(19, 19+14, 2343)-Net over F128 — Constructive and digital
Digital (19, 33, 2343)-net over F128, using
- 1 times m-reduction [i] based on digital (19, 34, 2343)-net over F128, using
- net defined by OOA [i] based on linear OOA(12834, 2343, F128, 15, 15) (dual of [(2343, 15), 35111, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(12834, 16402, F128, 15) (dual of [16402, 16368, 16]-code), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- linear OA(12829, 16385, F128, 15) (dual of [16385, 16356, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(12817, 16385, F128, 9) (dual of [16385, 16368, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 16385 | 1284−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(1285, 17, F128, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,128)), using
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- Reed–Solomon code RS(123,128) [i]
- discarding factors / shortening the dual code based on linear OA(1285, 128, F128, 5) (dual of [128, 123, 6]-code or 128-arc in PG(4,128)), using
- construction X applied to C([0,7]) ⊂ C([0,4]) [i] based on
- OOA 7-folding and stacking with additional row [i] based on linear OA(12834, 16402, F128, 15) (dual of [16402, 16368, 16]-code), using
- net defined by OOA [i] based on linear OOA(12834, 2343, F128, 15, 15) (dual of [(2343, 15), 35111, 16]-NRT-code), using
(19, 19+14, 9363)-Net in Base 128 — Constructive
(19, 33, 9363)-net in base 128, using
- 1281 times duplication [i] based on (18, 32, 9363)-net in base 128, using
- base change [i] based on digital (14, 28, 9363)-net over F256, using
- net defined by OOA [i] based on linear OOA(25628, 9363, F256, 14, 14) (dual of [(9363, 14), 131054, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- OA 7-folding and stacking [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- net defined by OOA [i] based on linear OOA(25628, 9363, F256, 14, 14) (dual of [(9363, 14), 131054, 15]-NRT-code), using
- base change [i] based on digital (14, 28, 9363)-net over F256, using
(19, 19+14, 16404)-Net over F128 — Digital
Digital (19, 33, 16404)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12833, 16404, F128, 14) (dual of [16404, 16371, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(6) [i] based on
- linear OA(12827, 16384, F128, 14) (dual of [16384, 16357, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(12813, 16384, F128, 7) (dual of [16384, 16371, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(1286, 20, F128, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,128)), using
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- Reed–Solomon code RS(122,128) [i]
- discarding factors / shortening the dual code based on linear OA(1286, 128, F128, 6) (dual of [128, 122, 7]-code or 128-arc in PG(5,128)), using
- construction X applied to Ce(13) ⊂ Ce(6) [i] based on
(19, 19+14, 18618)-Net in Base 128
(19, 33, 18618)-net in base 128, using
- 1281 times duplication [i] based on (18, 32, 18618)-net in base 128, using
- base change [i] based on digital (14, 28, 18618)-net over F256, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 18618, F256, 3, 14) (dual of [(18618, 3), 55826, 15]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 21847, F256, 3, 14) (dual of [(21847, 3), 65513, 15]-NRT-code), using
- OOA 3-folding [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(25627, 65536, F256, 14) (dual of [65536, 65509, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(25623, 65536, F256, 12) (dual of [65536, 65513, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- OOA 3-folding [i] based on linear OA(25628, 65541, F256, 14) (dual of [65541, 65513, 15]-code), using
- discarding factors / shortening the dual code based on linear OOA(25628, 21847, F256, 3, 14) (dual of [(21847, 3), 65513, 15]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25628, 18618, F256, 3, 14) (dual of [(18618, 3), 55826, 15]-NRT-code), using
- base change [i] based on digital (14, 28, 18618)-net over F256, using
(19, 19+14, large)-Net in Base 128 — Upper bound on s
There is no (19, 33, large)-net in base 128, because
- 12 times m-reduction [i] would yield (19, 21, large)-net in base 128, but