<s>
AN	B-Error_Name
codes	I-Error_Name
are	O
error-correcting	B-Error_Name
code	I-Error_Name
that	O
are	O
used	O
in	O
arithmetic	O
applications	O
.	O
</s>
<s>
AN	B-Error_Name
codes	I-Error_Name
are	O
arithmetic	O
codes	O
that	O
are	O
named	O
for	O
the	O
integers	O
and	O
that	O
are	O
used	O
to	O
encode	O
and	O
decode	O
the	O
codewords	O
.	O
</s>
<s>
As	O
with	O
codes	O
using	O
hamming	O
distance	O
,	O
AN	B-Error_Name
codes	I-Error_Name
can	O
correct	O
up	O
to	O
errors	O
where	O
is	O
the	O
distance	O
of	O
the	O
code	O
.	O
</s>
<s>
To	O
ensure	O
that	O
the	O
distance	O
of	O
the	O
code	O
will	O
not	O
be	O
too	O
small	O
,	O
we	O
will	O
define	O
modular	O
AN	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
By	O
using	O
modular	O
weight	O
with	O
,	O
the	O
AN	B-Error_Name
codes	I-Error_Name
will	O
be	O
cyclic	O
code	O
.	O
</s>
<s>
Cyclic	O
AN	B-Error_Name
codes	I-Error_Name
are	O
a	O
subset	O
of	O
cyclic	O
codes	O
and	O
have	O
the	O
same	O
properties	O
.	O
</s>
<s>
The	O
Mandelbaum-Barrows	O
Codes	O
are	O
a	O
type	O
of	O
cyclic	O
AN	B-Error_Name
codes	I-Error_Name
introduced	O
by	O
D	O
.	O
Mandelbaum	O
and	O
J	O
.	O
T	O
.	O
Barrows	O
.	O
</s>
