MinT - Best Known (120, 120+11, s)-Nets in Base 5

Best Known (120, 120+11, s)-Nets in Base 5

(120, 120+11, 3630842)-Net over F₅ — Constructive and digital

Digital (120, 131, 3630842)-net over F₅, using

generalized (u,Â u+v)-construction [i] based on
1. digital (11, 14, 976561)-net over F₅, using
  - s-reduction based on digital (11, 14, 1123200)-net over F₅, using
    - net defined by OOA [i] based on linear OOA(5¹⁴, 1123200, F₅, 3, 3) (dual of [(1123200, 3), 3369586, 4]-NRT-code), using
      - appending kth column [i] based on linear OOA(5¹⁴, 1123200, F₅, 2, 3) (dual of [(1123200, 2), 2246386, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(5¹⁴, 1123200, F₅, 3) (dual of [1123200, 1123186, 4]-code or 1123200-cap in PG(13,5)), using
        product of two projective caps and an affine cap avoiding two hyperplanes [i] based on
        linear OA(5⁴, 26, F₅, 3) (dual of [26, 22, 4]-code or 26-cap in PG(3,5)), using
        ovoid in PG(3,Â 5) [i]
        linear OA(5⁷, 675, F₅, 3) (dual of [675, 668, 4]-code or 675-cap in PG(6,5)), using
        product of two caps with tangent hyperplane [i]
2. digital (31, 36, 976561)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5³⁶, 976561, F₅, 5, 5) (dual of [(976561, 5), 4882769, 6]-NRT-code), using
    - OOA 2-folding and stacking with additional row [i] based on linear OA(5³⁶, 1953123, F₅, 5) (dual of [1953123, 1953087, 6]-code), using
      - discarding factors / shortening the dual code based on linear OA(5³⁶, 1953124, F₅, 5) (dual of [1953124, 1953088, 6]-code), using
        1 times truncation [i] based on linear OA(5³⁷, 1953125, F₅, 6) (dual of [1953125, 1953088, 7]-code), using
        an extension C_e(5) of the primitive narrow-sense BCH-code C(I) with length 1953124Â = 5⁹âˆ’1, defining interval IÂ =Â [1,5], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 6 [i]
3. digital (70, 81, 1677720)-net over F₅, using
  - net defined by OOA [i] based on linear OOA(5⁸¹, 1677720, F₅, 11, 11) (dual of [(1677720, 11), 18454839, 12]-NRT-code), using
    - OOA 5-folding and stacking with additional row [i] based on linear OA(5⁸¹, 8388601, F₅, 11) (dual of [8388601, 8388520, 12]-code), using
      - discarding factors / shortening the dual code based on linear OA(5⁸¹, large, F₅, 11) (dual of [large, large−81, 12]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 9765626Â | 5²⁰âˆ’1, defining interval IÂ =Â [0,5], and minimum distance dÂ â‰¥ |{−5,−4,…,5}|+1Â =Â 12 (BCH-bound) [i]

(120, 120+11, large)-Net over F₅ — Digital

Digital (120, 131, large)-net over F₅, using

5⁴ times duplication [i] based on digital (116, 127, large)-net over F₅, using
- t-expansion [i] based on digital (112, 127, large)-net over F₅, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(5¹²⁷, large, F₅, 15) (dual of [large, large−127, 16]-code), using
    - 7 times code embedding in larger space [i] based on linear OA(5¹²⁰, large, F₅, 15) (dual of [large, large−120, 16]-code), using
      - the primitive narrow-sense BCH-code C(I) with length 9765624Â = 5¹⁰âˆ’1, defining interval IÂ =Â [1,15], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(120, 120+11, large)-Net in Base 5 — Upper bound on s

There is no (120, 131, large)-net in base 5, because

9 times m-reduction [i] would yield (120, 122, large)-net in base 5, but
- mutually orthogonal hypercube bound [i]

Best Known (120, 120+11, s)-Nets in Base 5

(120, 120+11, 3630842)-Net over F5 — Constructive and digital

(120, 120+11, large)-Net over F5 — Digital

(120, 120+11, large)-Net in Base 5 — Upper bound on s

(120, 120+11, 3630842)-Net over F₅ — Constructive and digital

(120, 120+11, large)-Net over F₅ — Digital