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=Alternant code=
 
A class of parameterised [[Error detection and correction|error-correcting codes]] which generalise the [[BCH code]]s.
 
 
An ''alternant code'' over $GF(q)$ of length $n$ is defined by a parity check matrix $H$ of [[alternant matrix|alternant]] form $H_{i,j} = \alpha_j^i y_i$, where the $\alpha_j$ are distinct elements of the extension $GF(q^m)$, the $y_i$ are further non-zero parameters again in the extension $GF(q^m)$ and the indices range as $i$ from 0 to $\delta-1$, $j$ from 1 to $n$.
 
 
The parameters of this alternant code are length $n$, dimension $\ge n - m\delta$ and minimum distance $\ge \delta+1$.
 
There exist long alternant codes which meet the [[Gilbert-Varshamov bound]].
 
 
The class of alternant codes includes [[BCH code]]s, [[Goppa code]]s and [[Srivasta code]]s.
 
 
== References ==
 
* {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | date=1977 | isbn=0-444-85193-3 | pages=332-338 }}
 
  
 
=Szpiro's conjecture=
 
=Szpiro's conjecture=

Revision as of 06:24, 7 September 2013




Szpiro's conjecture

A conjectural relationship between the conductor and the discriminant of an elliptic curve. In a general form, it is equivalent to the well-known ABC conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.

The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have

\[ \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. \, \]

The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f, we have

\[ \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, \]

References

Wiener–Ikehara theorem

A Tauberian theorem relating the behaviour of a real sequence to the analytic properties of the associated Dirichlet series. It is used in the study of arithmetic functions and yields a proof of the Prime number theorem. It was proved by Norbert Wiener and his student Shikao Ikehara in 1932.


Let $F(x)$ be a non-negative, monotonic decreasing function of the positive real variable $x$. Suppose that the Laplace transform $$ \int_0^\infty F(x)\exp(-xs) dx $$ converges for $\Re s >1$ to the function $f(s)$ and that $f(s)$ is analytic for $\Re s \ge 1$, except for a simple pole at $s=1$ with residue 1. Then the limit as $x$ goes to infinity of $e^{-x} F(x)$ is equal to 1.

An important number-theoretic application of the theorem is to Dirichlet series of the form $\sum_{n=1}^\infty a(n) n^{-s}$ where $a(n)$ is non-negative. If the series converges to an analytic function in $\Re s \ge b$ with a simple pole of residue $c$ at $s = b$, then $\sum_{n\le X}a(n) \sim c \cdot X^b$.

Applying this to the logarithmic derivative of the Riemann zeta function, where the coefficients in the Dirichlet series are values of the von Mangoldt function, it is possible to deduce the prime number theorem from the fact that the zeta function has no zeroes on the line $\Re (s)=1$.

References

How to Cite This Entry:
Richard Pinch/sandbox-CZ. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-CZ&oldid=30392