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```Succinct representation
of codes with
applications to testing
Elena Grigorescu
Tali Kaufman
Outline
► Testing membership in error correcting codes
► Sufficient conditions for testing algebraic codes
► Possible promising perspective: rich group of
symmetries of code
► Our result: affine/cyclic invariant, sparse codes can
be described succinctly by a single, short
codeword
► Implies locally testability results
► Proof sketch
► Conclusions
Locally testable codes
f : FN  F2 , N  2
0 1
n
1 …
0
CC
q queries
C satisfies
Code:  ( f , g )   0
Linear:
f , g C
f C
-Accept w.p 1 if
-Reject w.p. ε if  ( f , C )   1
(
q
independent of n)
Testing linear codes via duality
► dual
 C  { y  { 0, 1} :  y, x  0,  x  C }
N
► [BHR]
Test for linear properties are essentially
of the form:
1. Given x, pick
2. Accept iff  y, x  0
of test: weight ( y)  | {i | yi  0} |
► Dual-distance: smallest weight of a codeword in
dual-C
► Locality
Sufficient conditions for testing



Necessary condition for local testing (linear codes):
- small “dual distance”
- not sufficient( [BHR] show random LDPC not
locally testable)
Sufficient conditions
- Possible approach: nice symmetries of code
C is invariant under permutation  : [ N ]  [ N ] iff
x  ( x1 , x2 ,..., x N )  C  x  ( x (1) , x ( 2) ,..., x ( N ) )  C
 Aut (C )  { : x  C , x  C}
Symmetries and testing
Many known testable codes have somewhat large symmetry groups:
Eg. Linearity: invariance under general linear group
Low degree, Reed-Muller, BCH: invariance under affine group
Specific sufficient condition:
[KS] affine invariance + ‘local characterization’ imply testing
AKKLR Conjecture: 2 transitivity + small dual distance
Falsified in general [GKS]
Modified AKKLR Question: What if dual code is generated by single
low-weight codeword and its shifts under some group G (“SingleOrbit Property under G”)
Are these codes testable (for some group G? for all groups G?)
Single orbit property under affine
invariant/cyclic groups
►
►
Affine group:
Cyclic group:
{ : F2n  F2n |  ( x)  ax  b}
{ : F2n

 F2n

|  ( x)  ax}
C has single orbit under cyclic group:
w=01001 then B={01001, 10100, 01010, 00101, 10010} is a basis for C
►
►
Formally, C has k-single orbit under G ( included in Aut(C) ) if
c  C , wt(c)  k
s.t.
C  Span ({c   |   G})
Our work
►
Study “Single-Orbit Property” of common codes.
►
Def: C is sparse if it contains a poly number of codewords
►
Duals of binary sparse + affine invariant codes have the single-orbit
property under affine group
- under some block-length restriction: n prime
- [KS’08] Single-orbit codes under affine group are testable.
►
Duals of binary sparse + cyclic invariant codes have the single-orbit
property under cyclic group
- under more block-length restrictions: n, N-1 primes
- No testing implications
Related works
►
►
Sparse, large distance codes are testable
[KL, KS] ( tests are coarse, unstructured)
Affine/linear invariant + “characterization” imply testing
►
Here: sparse
affine invariance
►
[KL] dual-e-BCH codes are testable (unstructured tests)
e-BCH are spanned by shortest codewords
►
►
large distance
“characterization” (explicit tests)
Here: dual-e-BCH are spanned by a single, short codeword (explicit
basis / tests)
Toward an explicit description of binary affine
invariant codes
 : F2  F2
 ( x)  ax  b
►
Affine invariance:
►
Any function f : F  F2 is of the form
2
n
n
n
f ( x)  Trace ( p( x))
►
The Trace function: Trace ( x)  x  x 2  x 2    x 2
2
n 1
Trace : F2n  F2
Trace( x  y )  Trace( x)  Trace( y)
Trace(x )  Trace(x )  ...  Trace(x
i
2i
2
n -1
i
)
Explicit description of sparse affine families
►
Let
►
- What aff inv families does f belong to?
Consider the binary rep of degrees: 1, 111, 1100, 10011
Trace( )  C
 Then
f ( x)  Trace( x  x  x
7
Trace( x
101
)C
Trace( x
011
)C
Trace( x
10001
Trace(x
►
►
►
12
x )
19
)C
10001
 x
011
)  C , etc...
In general: if degree d occurs then its shadow occurs
Sparsity
translates= into
few monomials
{10011,10010,10001,10000,11,10,1}
Affine/Cyclic codes are described by a small set of degrees
Proof ingredients

Strong number theoretic result of Bourgain implies
high weight of functions of the form
Trace(a1 x
d1
 ...  a k x
few degs >
Degs
inside
trace
0
2
2
 g ( x))
dk
deg< 2 n / 2
n/2
n/2
2
n (1 )
2 1
n
?
Weil bounds
Bourgain
Proof ingredients (contd)
MacWilliams type counting estimates
- fourier transform between the functions that represent
number of codewords for each weight in C and in dual- C,
respectively


For sparse codes of length N and of high distance obtain:
# codew. weight k in dual - C 
N
k
|C|
Proof sketch
Want: exists codew. c with wt < k s.t.
Span(aff(c))=Dual-C
Dual-C
C’
►
►
►
►
C(a)
w
C
Dual-C’
►
►
►
►
weight<k
C described by set of degrees D
Let dual-C’= Span( aff(w) )
If C’
C then there exists a  D
a
Trace( x )  C ' / C
Let C(a)  Trace( x a )  C
Associate C(a) to codew. w
Does every wt<k codew. belong
to a dual of some C(a) ?
New goal: exists w that does
not belong to the dual of any
C(a), for all a
We show
| (dual - C) k |  | ( dual - C(a) ) k |
Proof Sketch
►
►
►
C, C(a): sparse, high dist (Bourgain) (assuming N-1 and n
are primes)
How many codew of wt k in dual-C?
N
k
N
t
How many codew of wt k in dual-C(a) ?
N
N
k
t 1
►
Total number of degrees a to consider: N/n
►
Therefore, there exists codew. of wt<k in dual-C that whose
orbit generates C
Specifics of the affine case proof
► Here only assume n prime- Bourgain doesn’t hold
for all monomials
► Need codes C(a) to have deg a < N 1
►
Use shadow property Trace ( x 11101111)  C' 
Trace(x
►
)  C'
10000101
Show that enough to consider a in the set
{2  1 | i  [n] }
i
Cyclic codes

under:  : F2  F2  ( x)  ax
► Punctured affine invariant codes are cyclic
► Cyclic codes are described by generator
polynomial (or its roots in the field)
► Alternatively described by function families of
the form
► Invariant
n
n
C  {Trace ( ai x )}
i
i 0
► Degrees
can be arbitrary
Single orbit: affine vs cyclic codes
►
Affine (length N= 2 n )
 n prime
 degrees of monomials
 |Aut(C)|= N 2
 “single orbit” implies
testing
► Cyclic
(length N-1)
 n, N-1 primes
 degrees of monomials
are arbitrary
 |Aut(C)|=N
 not known if “single
orbit” implies testing
Open Questions
►
Do same results hold for non-prime n, 2 n  1 ?
►
Single orbit under what other groups imply testing?
How large does the Aut group should be to imply
testing?
►
Small weight basis + invariance implies testing?
►
Examples of families where the tests are not the
“expected” ones (I.e. not the ones suggested by the
description of Aut group)
Thank you
```
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