M algebraic-coding-theory-notes.pdf → algebraic-coding-theory-notes.pdf

M algebraic-coding-theory-notes.tex → algebraic-coding-theory-notes.tex

@@ -1299,7 +1299,7 @@ \item correct up to $t$ errors in any codeword if $d(C) \geq 2t + 1$
 
       This is, the spheres of radius $t$ centered on the codewords of $C$ are disjoint (ie. have no overlap).
 
-      So, if $\leq t$ errors occur, then the received vector $y$ may be different from the sent codeword $x$, hence being different from the centre of the sphere $S(x, t)$, but can not 'escape' from the sphere, so is 'drawn back' to $x$ by nearest neigbour decoding.
+      So, if $\leq t$ errors occur, then the received vector $y$ may be different from the sent codeword $x$, hence being different from the centre of the sphere $S(x, t)$, but can not 'escape' from the sphere, so is 'drawn back' to $x$ by nearest neighbour decoding.
   \end{enumerate}
 
 
@@ -1436,7 +1436,7 @@
   So if $a=(a_1, a_2, \ldots, a_r) \in P$, then $P= \{ \lambda a ~|~ \lambda \in GF(q) \}$.
 
   $\Longrightarrow$ for $a=(a_1, a_2, \ldots, a_r),~b=(b_1, b_2, \ldots, b_r) \in P$, then
-  $$a=b ~\text{in}}~ PG(r-1,q) ~\text{iff}~ a=\lambda b ~\text{in}~ V(r,q)$$
+  $$a=b ~\text{in}~ PG(r-1,q) ~\text{iff}~ a=\lambda b ~\text{in}~ V(r,q)$$
   for some non-zero scalar $\lambda$.
 \end{defn}