The Sphere Packing Bound via Augustin’s Method

Abstract

Canım halam Fatma Nakiboglu Aydiç'in anısına adanmıştır. Dedicated to the memory of my dear aunt Fatma Nakiboglu Aydiç.Abstract-A sphere packing bound (SPB) with a prefactor that is polynomial in the block length n is established for codes on a length n product channel W [1,n] assuming that the maximum order ½ Rényi capacity among the component channels, i.e. max t∈[1,n] C1 /2,Wt , is O(ln n). The reliability function of the discrete stationary product channels with feedback is bounded from above by the sphere packing exponent. Both results are proved by first establishing a non-asymptotic SPB. The latter result continues to hold under a milder stationarity hypothesis.

The Sphere Packing Bound for DSPCs With Feedback à la Augustin

“…Recently, we have proposed another proof for the SPB for codes on DSPCs with feedback [11,Thm 3] and generalized it to codes on (possibly non-stationary) DPCs with feedback [11,Thm 4]. It seems analogous generalizations are possible for Proposition 1 and Lemma 8 under similar hypotheses.…”

“…In a general stationary product channel with feedback, the stochastic matrix W ∈ P(Y|X) is replaced by a transition probability W ∈ P(Y|X ), see [11,Definition 8]. In order to generalize Lemma 8 to stationary product channels with feedback, we first need to prove Lemma 4.…”

“…In [10], Augustin provided the first proof of the SPB on the product channels that does not assume either the stationarity of the channel or the finiteness of its input set. In [11], we have improved the approximation error term of the upper bound on the error exponent given in [10] from O n −0.5 to O n −1 ln n for the block length n, using the Rényi capacity and center analyzed in [12].…”

The Augustin center and the sphere packing bound for memoryless channels

“…Proofs of the SPB based on Augustin's method are exceptions to this observation: [24]- [26] do not assume either the finiteness of the input set or a specific noise structure, nor do they assume the stationarity of the channel. However, [24], [25, §31], [26] establish the SPB for the product channels, rather than the memoryless channels; hence proofs of the SPB for the composition constrained codes 1 on the stationary channels [9]- [20] -which include the important special case of the cost constrained ones [15]- [20]-are not subsumed by [24], [25, §31], or [26]. In [25, §36], Augustin proved the SPB for the cost constrained (possibly non-stationary) memoryless channels assuming a bounded cost function.…”

“…Theorem 2, presented in §3, establishes the SPB for a framework that subsumes all of the models considered in [2]- [26] by employing [27], which analyzes Augustin's information measures. Our use of [27] and Augustin's information measures is similar to the use of [28] and Rényi 's information measures in [26]. For the product channels, [26,Thm.…”