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So I am working with cyclic codes of length $n=15$ over $\mathbb{F}_{2}$.

I have all my cyclotomic cosets modulo $15$: $C_0=\{0\}$, $C_1=\{1,2,4,8\}$, $C_3=\{3,6,12,9\}$, $C_5=\{5,10\}$, $C_7=\{7,14,13,11\}$

So I am trying to figure out properties of all $2^5=32$ codes. I am a bit stuck on determining if a code is a BCH code. My text gives the following definition:

Let $\delta$ by an integer with $2\leq \delta \leq n$. A BCH code $C$ over $\mathbb{F}_q$ of length $n$ and designed distance $\delta$ is a cyclic code with defining set $T=C_b \cup C_{b+1} \cup ... \cup C_{b+\delta-2}$, where $C_i$ is the $q$-cyclotomic coset modulo $n$ containing $i$.

I am a bit unsure of to apply this.

For example, let $T=C_1$. Then this would be BCH with $b=1$ and $\delta=2$. Is this correct?

What about $T=C_1 \cup C_3 \ C_5$ then this is the same as $T=C_1 \cup C_2 \cup C_3 \cup C_4 \cup C_5 \cup C_6$. So this is BCH with $b=1$ and $\delta=7$ as $b+\delta-2=1+7-2=6$. Would this be correct?

Next, what about finding the largest $\delta$ and the Bose distance?

Thanks for the help.

user3784030
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1 Answers1

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Yes, if $T=C_1$ then $\{1,2\}$ are consecutive and you get $b=1,\delta=1.$

Note, if $T=C_1\cup C_3$, then you get $T=\{1,2,3,4,6,8,9\}$ so you can take $b=1,\delta=3.$

Further, if $T=C_1\cup C_3 \cup C_5,$ then $T=\{1,2,3,4,5,6,8,9,10\}$ which gives $b=1,\delta=7.$ However the notation you have below $$T=C_1 \cup C_2 \cup C_3 \cup C_4 \cup C_5 \cup C_6$$ is redundant! $C_1=C_2=C_4$ and $C_3=C_6$. Maybe this is the confusion.

Some people say "generalized BCH" if $b\neq 1$, some don't. Some people reserve that for the case when you don't take primitive elements and powers, which you do here. In general, a BCH code is any code where the so called ''null spectrum'' or the set of zeroes is a union of cyclotomic cosets.

I don't know what Bose distance is. I don't think it's standard terminology. But note that the BCH bound says the actual minimum distance is at least $\delta$ if $b,b+1,\ldots,b+\delta-2$ are zeroes of a BCH code. There are generalizations of this due to Roos, and Hartmann-Tzeng which mean that sometimes you might get a higher distance than $\delta$ if some arithmetic progresssion structure to the zeroes exist.

By the way, the Wikipedia page on BCH codes seems to be quite good.

kodlu
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  • Hi!!! Could you take a look at my question: http://math.stackexchange.com/questions/1759898/distance-of-bch-code ? – Evinda Apr 26 '16 at 21:48