Best Known (7, 13, s)-Nets in Base 128
(7, 13, 5464)-Net over F128 — Constructive and digital
Digital (7, 13, 5464)-net over F128, using
- net defined by OOA [i] based on linear OOA(12813, 5464, F128, 6, 6) (dual of [(5464, 6), 32771, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- OA 3-folding and stacking [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
(7, 13, 16392)-Net over F128 — Digital
Digital (7, 13, 16392)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(12813, 16392, F128, 6) (dual of [16392, 16379, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- linear OA(12811, 16384, F128, 6) (dual of [16384, 16373, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1285, 16384, F128, 3) (dual of [16384, 16379, 4]-code or 16384-cap in PG(4,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1282, 8, F128, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,128)), using
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- Reed–Solomon code RS(126,128) [i]
- discarding factors / shortening the dual code based on linear OA(1282, 128, F128, 2) (dual of [128, 126, 3]-code or 128-arc in PG(1,128)), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
(7, 13, 21846)-Net in Base 128 — Constructive
(7, 13, 21846)-net in base 128, using
- net defined by OOA [i] based on OOA(12813, 21846, S128, 6, 6), using
- OA 3-folding and stacking [i] based on OA(12813, 65538, S128, 6), using
- discarding parts of the base [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding parts of the base [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- OA 3-folding and stacking [i] based on OA(12813, 65538, S128, 6), using
(7, 13, 32768)-Net in Base 128
(7, 13, 32768)-net in base 128, using
- net defined by OOA [i] based on OOA(12813, 32768, S128, 9, 6), using
- OOA stacking with additional row [i] based on OOA(12813, 32769, S128, 3, 6), using
- discarding parts of the base [i] based on linear OOA(25611, 32769, F256, 3, 6) (dual of [(32769, 3), 98296, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25611, 32769, F256, 2, 6) (dual of [(32769, 2), 65527, 7]-NRT-code), using
- OOA 2-folding [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(25611, 65536, F256, 6) (dual of [65536, 65525, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2569, 65536, F256, 5) (dual of [65536, 65527, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- OOA 2-folding [i] based on linear OA(25611, 65538, F256, 6) (dual of [65538, 65527, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(25611, 32769, F256, 2, 6) (dual of [(32769, 2), 65527, 7]-NRT-code), using
- discarding parts of the base [i] based on linear OOA(25611, 32769, F256, 3, 6) (dual of [(32769, 3), 98296, 7]-NRT-code), using
- OOA stacking with additional row [i] based on OOA(12813, 32769, S128, 3, 6), using
(7, 13, large)-Net in Base 128 — Upper bound on s
There is no (7, 13, large)-net in base 128, because
- 4 times m-reduction [i] would yield (7, 9, large)-net in base 128, but