A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently [Y] introduced a novel technique for constructing locally decodable codes and vastly improved the upper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work of [Y] and argue that further progress via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m=2^t-1 that has a prime factor p>m^\gamma yields a family of k(\gamma)-query locally decodable codes of length Exp(n^{1/t}). Conversely, if for some fixed k and all \epsilon > 0 one can use the technique of [Y] to obtain a family of k-query LDCs of length Exp(n^\epsilon); then infinitely many Mersenne numbers have prime factors arger than known currently.
Deep Dive into Locally Decodable Codes From Nice Subsets of Finite Fields and Prime Factors of Mersenne Numbers.
A k-query Locally Decodable Code (LDC) encodes an n-bit message x as an N-bit codeword C(x), such that one can probabilistically recover any bit x_i of the message by querying only k bits of the codeword C(x), even after some constant fraction of codeword bits has been corrupted. The major goal of LDC related research is to establish the optimal trade-off between length and query complexity of such codes. Recently [Y] introduced a novel technique for constructing locally decodable codes and vastly improved the upper bounds for code length. The technique is based on Mersenne primes. In this paper we extend the work of [Y] and argue that further progress via these methods is tied to progress on an old number theory question regarding the size of the largest prime factors of Mersenne numbers. Specifically, we show that every Mersenne number m=2^t-1 that has a prime factor p>m^\gamma yields a family of k(\gamma)-query locally decodable codes of length Exp(n^{1/t}). Conversely, if for s
Classical error-correcting codes allow one to encode an n-bit string x into in N -bit codeword C(x), in such a way that x can still be recovered even if C(x) gets corrupted in a number of coordinates. It is well-known that codewords C(x) of length N = O(n) already suffice to correct errors in up to δN locations of C(x) for any constant δ < 1/4. The disadvantage of classical error-correction is that one needs to consider all or most of the (corrupted) codeword to recover anything about x. Now suppose that one is only interested in recovering one or a few bits of x. In such case more efficient schemes are possible. Such schemes are known as locally decodable codes (LDCs). Locally decodable codes allow reconstruction of an arbitrary bit x i , from looking only at k randomly chosen coordinates of C(x), where k can be as small as 2. Locally decodable codes have numerous applications in complexity theory [15,29], cryptography [6,11] and the theory of fault tolerant computation [24]. Below is a slightly informal definition of LDCs:
A (k, δ, ǫ)-locally decodable code encodes n-bit strings to N -bit codewords C(x), such that for every i ∈ [n], the bit x i can be recovered with probability 1 -ǫ, by a randomized decoding procedure that makes only k queries, even if the codeword C(x) is corrupted in up to δN locations.
One should think of δ > 0 and ǫ < 1/2 as constants. The main parameters of interest in LDCs are the length N and the query complexity k. Ideally we would like to have both of them as small as possible. The concept of locally decodable codes was explicitly discussed in various papers in the early 1990s [2,28,21]. Katz and Trevisan [15] were the first to provide a formal definition of LDCs. Further work on locally decodable codes includes [3,8,20,4,16,30,34,33,14,23].
Below is a brief summary of what was known regarding the length of LDCs prior to [34]. The length of optimal 2-query LDCs was settled by Kerenidis and de Wolf in [16] and is exp(n). 1 The best upper bound for the length of 3-query LDCs was exp n 1/2 due to Beimel et al. [3], and the best lower bound is Ω(n 2 ) [33]. For general (constant) k the best upper bound was exp n O(log log k/(k log k)) due to Beimel et al. [4] and the best lower bound is Ω n 1+1/(⌈k/2⌉-1) [33].
The recent work [34] improved the upper bounds to the extent that it changed the common perception of what may be achievable [12,11]. [34] introduced a novel technique to construct codes from so-called nice subsets of finite fields and showed that every Mersenne prime p = 2 t -1 yields a family of 3-query LDCs of length exp n 1/t . Based on the largest known Mersenne prime [9], this translates to a length of less than exp n 10 -7 . Combined with the recursive construction from [4], this result yields vast improvements for all values of k > 2. It has often been conjectured that the number of Mersenne primes is infinite. If indeed this conjecture holds, [34] gets three query locally decodable codes of length N = exp n O "
for infinitely many n. Finally, assuming that the conjecture of Lenstra, Pomerance and Wagstaff [31,22,32] regarding the density of Mersenne primes holds, [34] gets three query locally decodable codes of length N = exp n O "
for all n, for every ǫ > 0.
In this paper we address two natural questions left open by [34]:
- Are Mersenne primes necessary for the constructions of [34]? 2. Has the technique of [34] been pushed to its limits, or one can construct better codes through a more clever choice of nice subsets of finite fields?
We extend the work of [34] and answer both of the questions above. In what follows let P (m) denote the largest prime factor of m. We show that one does not necessarily need to use Mersenne primes. It suffices to have Mersenne numbers with polynomially large prime factors. Specifically, every Mersenne number m = 2 t -1 such that P (m) ≥ m γ yields a family of k(γ)-query locally decodable codes of length exp n 1/t . A partial converse also holds. Namely, if for some fixed k ≥ 3 and all ǫ > 0 one can use the technique of [34] to (unconditionally) obtain a family of k-query LDCs of length exp (n ǫ ) ; then for infinitely many t we have
The bound (1) may seem quite weak in light of the widely accepted conjecture saying that the number of Mersenne primes is infinite. However (for any k ≥ 3) this bound is substantially stronger than what is currently known unconditionally. Lower bounds for P (2 t -1) have received a considerable amount of attention in the number theory literature [25,26,10,27,19,18]. The strongest result to date is due to Stewart [27]. It says that for all integers t ignoring a set of asymptotic density zero, and for all functions ǫ(t) > 0 where ǫ(t) tends to zero monotonically and arbitrarily slowly:
(2)
1 Throughout the paper we use the standard notation exp(x)
There are no better bounds known to hold for infinitely many values of t, unless one is willing to accept some number theoretic conjectures [19,18]. We
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