Codes for Partially Stuck-At Memory Cells

Abstract

In this work, we study a new model of defect memory cells, called partially stuck-at memory cells, which is motivated by the behavior of multi-level cells in non-volatile memories such as flash memories and phase change memories. If a cell can store the q levels 0, 1, . . . , q − 1, we say that it is partially stuck-at level s, where 1 ≤ s ≤ q − 1, if it can only store values which are at least s. We follow the common setup where the encoder knows the positions and levels of the partially stuck-at cells whereas the decoder does not.Our main contribution in the paper is the study of codes for masking u partially stuck-at cells. We first derive lower and upper bounds on the redundancy of such codes. The upper bounds are based on two trivial constructions. We then present three code constructions over an alphabet of size q, by first considering the case where the cells are partially stuck-at level s = 1. The first construction works for u < q and is asymptotically optimal if u + 1 divides q. The second construction uses the reduced row Echelon form of matrices to generate codes for the case u ≥ q, and the third construction solves the case of arbitrary u by using codes which mask binary stuck-at cells. We then show how to generalize all constructions to arbitrary stuck levels. Furthermore, we study the dual defect model in which cells cannot reach higher levels, and show that codes for partially stuck-at cells can be used to mask this type of defects as well. Lastly, we analyze the capacity of the partially stuck-at memory channel and study how far our constructions are from the capacity.

Error Correction for Partially Stuck Memory Cells

“…1 An asymptotically optimal class of codes was proposed to correct ρ defects (ρ = const value) regardless of n → ∞, where n is the code length. 2 Asymptotically optimal linear codes were proposed to correct ρ defects (ρ = αn), where α = constant value and ρ = number of defects that is linearly increasing while n → ∞, i.e. if n is doubled then ρ is doubled.…”

“…A cell is called stuck at level s, where s ∈ [q], if it cannot change its value and always stores the value s. A cell is called partially stuck at level s, where s ∈ [q], if it can only store values which are at least s. If a cell is partially stuck at 0, it is a non-defect cell which can store any of the q levels [2].…”

“…The goal of this paper is to find a code construction that can store information in a memory with some partially stuck cells and additionally can correct errors. For the masking process and according to [2], we need only a single redundancy symbol if u < q and (s φi = 1). However, the work in [2] does not consider correcting additional random errors.…”

“…For multi-level PCMs, the case s = 1 is particularly important since this means that a cell cannot reach the amorphous state anymore, but all partially crystalline ones, cf. [2]. Similarly, (partially) stuck cells can occur in flash This work has received funding from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Grant No.…”

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We present code constructions for masking u partially stuck memory cells with q levels and correcting additional random errors. The results are achieved by combining the methods for masking and error correction for stuck cells in [1] with the masking-only results for partially stuck cells in [2]. We present two constructions for masking u < q cells and error correction: one is general and based on a generator matrix of a specific form.

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