MinT - Best Known (116−15, 116, s)-Nets in Base 16

Best Known (116−15, 116, s)-Nets in Base 16

(116−15, 116, 2440690)-Net over F₁₆ — Constructive and digital

Digital (101, 116, 2440690)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (23, 30, 43948)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (1, 4, 257)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16⁴, 257, F₁₆, 3, 3) (dual of [(257, 3), 767, 4]-NRT-code), using
        appending kth column [i] based on linear OOA(16⁴, 257, F₁₆, 2, 3) (dual of [(257, 2), 510, 4]-NRT-code), using
        OAs with strength 3, bÂ â‰ Â 2, and mÂ >Â 3 are always embeddable [i] based on linear OA(16⁴, 257, F₁₆, 3) (dual of [257, 253, 4]-code or 257-cap in PG(3,16)), using
        ovoid in PG(3,Â 16) [i]
    2. digital (19, 26, 43691)-net over F₁₆, using
      - net defined by OOA [i] based on linear OOA(16²⁶, 43691, F₁₆, 7, 7) (dual of [(43691, 7), 305811, 8]-NRT-code), using
        OOA 3-folding and stacking with additional row [i] based on linear OA(16²⁶, 131074, F₁₆, 7) (dual of [131074, 131048, 8]-code), using
        trace code [i] based on linear OA(256¹³, 65537, F₂₅₆, 7) (dual of [65537, 65524, 8]-code), using
        the expurgated narrow-sense BCH-code C(I) with length 65537Â | 256⁴âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
2. digital (71, 86, 2396742)-net over F₁₆, using
  - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(116−15, 116, 2484126)-Net in Base 16 — Constructive

(101, 116, 2484126)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (23, 30, 87384)-net in base 16, using
  - base change [i] based on digital (13, 20, 87384)-net over F₆₄, using
    - net defined by OOA [i] based on linear OOA(64²⁰, 87384, F₆₄, 7, 7) (dual of [(87384, 7), 611668, 8]-NRT-code), using
      - OOA 3-folding and stacking with additional row [i] based on linear OA(64²⁰, 262153, F₆₄, 7) (dual of [262153, 262133, 8]-code), using
        constructionÂ X4 applied to C([0,3]) âŠ‚ C([0,2]) [i] based on
        linear OA(64¹⁹, 262145, F₆₄, 7) (dual of [262145, 262126, 8]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,3], and minimum distance dÂ â‰¥ |{−3,−2,…,3}|+1Â =Â 8 (BCH-bound) [i]
        linear OA(64¹³, 262145, F₆₄, 5) (dual of [262145, 262132, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,2], and minimum distance dÂ â‰¥ |{−2,−1,0,1,2}|+1Â =Â 6 (BCH-bound) [i]
        linear OA(64⁷, 8, F₆₄, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,64)), using
        dual of repetition code with length 8 [i]
        linear OA(64¹, 8, F₆₄, 1) (dual of [8, 7, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, 64, F₆₄, 1) (dual of [64, 63, 2]-code), using
        Reedâ€“Solomon code RS(63,64) [i]
2. digital (71, 86, 2396742)-net over F₁₆, using
  - trace code for nets [i] based on digital (28, 43, 1198371)-net over F₂₅₆, using
    - net defined by OOA [i] based on linear OOA(256⁴³, 1198371, F₂₅₆, 15, 15) (dual of [(1198371, 15), 17975522, 16]-NRT-code), using
      - OOA 7-folding and stacking with additional row [i] based on linear OA(256⁴³, 8388598, F₂₅₆, 15) (dual of [8388598, 8388555, 16]-code), using
        discarding factors / shortening the dual code based on linear OA(256⁴³, large, F₂₅₆, 15) (dual of [large, large−43, 16]-code), using
        the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 256³âˆ’1, defining interval IÂ =Â [0,14], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 16 [i]

(116−15, 116, large)-Net over F₁₆ — Digital

Digital (101, 116, large)-net over F₁₆, using

t-expansion [i] based on digital (99, 116, large)-net over F₁₆, using
- 5 times m-reduction [i] based on digital (99, 121, large)-net over F₁₆, using
  - embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(16¹²¹, large, F₁₆, 22) (dual of [large, large−121, 23]-code), using
    - the primitive expurgated narrow-sense BCH-code C(I) with length 16777215Â = 16⁶âˆ’1, defining interval IÂ =Â [0,21], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 23 [i]

(116−15, 116, large)-Net in Base 16 — Upper bound on s

There is no (101, 116, large)-net in base 16, because

13 times m-reduction [i] would yield (101, 103, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]

Best Known (116−15, 116, s)-Nets in Base 16

(116−15, 116, 2440690)-Net over F16 — Constructive and digital

(116−15, 116, 2484126)-Net in Base 16 — Constructive

(116−15, 116, large)-Net over F16 — Digital

(116−15, 116, large)-Net in Base 16 — Upper bound on s

(116−15, 116, 2440690)-Net over F₁₆ — Constructive and digital

(116−15, 116, large)-Net over F₁₆ — Digital