Majority-logic Decoding with Subspace Designs

TL;DR

This paper explores majority-logic decoding for codes derived from subspace designs, showing they match the error correction capabilities of geometric design codes but with potentially much simpler decoding complexity.

Contribution

It introduces the use of subspace designs for majority-logic decoding, demonstrating comparable error correction with reduced decoding complexity.

Findings

Codes from subspace designs have the same decoding capability as geometric design codes.

Decoding complexity can be significantly reduced using subspace design codes.

The approach extends Rudolph's and Peterson-Weldon's decoding methods.

Abstract

Rudolph (1967) introduced one-step majority logic decoding for linear codes derived from combinatorial designs. The decoder is easily realizable in hardware and requires that the dual code has to contain the blocks of so called geometric designs as codewords. Peterson and Weldon (1972) extended Rudolphs algorithm to a two-step majority logic decoder correcting the same number of errors than Reed's celebrated multi-step majority logic decoder. Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved.

Tables

Table 1. Table 1: Code parameters from the construction in Theorem 4 a) for known subspace designs over 𝔽 2 subscript 𝔽 2 \mathbb{F}_{2} .

$t$-${(v,k,{\lambda }_{\text{known}})}_2$	${\lambda }_{\min }$	${\lambda }_{\max }$	${[n,\text{dim},\ell ]}_2$	$r$	$r_{\max }/r$	cyclic
$2$-${(3,2,1)}_2$	$1$	$1$	$[7,3,1]$	$3$		c
$2$-${(4,2,1)}_2$	$1$	$1$	$[15,4,3]$	$7$		c
$2$-${(4,3,3)}_2$	$3$	$3$	$[15,10,1]$	$7$		c
$2$-${(5,2,1)}_2$	$1$	$1$	$[31,5,7]$	$15$		c
$2$-${(5,3,7)}_2$	$7$	$7$	$[31,15,2]$	$35$		c
$2$-${(5,4,7)}_2$	$7$	$7$	$[31,25,1]$	$15$		c
$2$-${(6,2,1)}_2$	$1$	$1$	$[63,6,15]$	$31$		c
$2$-${(6,3,3)}_2$	$3$	$15$	$[63,21,5]$	$31$	$5.0$
$2$-${(6,4,35)}_2$	$35$	$35$	$[63,41,2]$	$155$		c
$2$-${(6,5,15)}_2$	$15$	$15$	$[63,56,1]$	$31$		c
$2$-${(7,2,1)}_2$	$1$	$1$	$[127,7,31]$	$63$		c
$2$-${(7,3,3)}_2$	$1$	$31$	$[127,28,10]$	$63$	$10.3$	c
$2$-${(7,4,15)}_2$	$5$	$155$	$[127,63,4]$	$135$	$10.3$	c
$2$-${(7,5,155)}_2$	$155$	$155$	$[127,98,2]$	$651$		c
$2$-${(7,6,31)}_2$	$31$	$31$	$[127,119,1]$	$63$		c
$2$-${(8,2,1)}_2$	$1$	$1$	$[255,8,63]$	$127$		c
$2$-${(8,3,21)}_2$	$21$	$63$	$[255,36,21]$	$889$	$3.0$	c
$2$-${(8,4,7)}_2$	$7$	$651$	$[255,92,9]$	$127$	$93.0$
$2$-${(8,5,465)}_2$	$465$	$1395$	$[255,162,4]$	$3937$	$3.0$	c
$2$-${(8,6,651)}_2$	$651$	$651$	$[255,218,2]$	$2667$		c
$2$-${(8,7,63)}_2$	$63$	$63$	$[255,246,1]$	$127$		c
$2$-${(9,2,1)}_2$	$1$	$1$	$[511,9,127]$	$255$		c
$2$-${(9,3,7)}_2$	$1$	$127$	$[511,45,42]$	$595$	$18.1$
$2$-${(9,4,21)}_2$	$7$	$2667$	$[511,129,18]$	$765$	$127.0$	c
$2$-${(9,5,93)}_2$	$31$	$11811$	$[511,255,8]$	$1581$	$127.0$	c
$2$-${(9,6,651)}_2$	$93$	$11811$	$[511,381,4]$	$5355$	$18.1$
$2$-${(9,8,127)}_2$	$127$	$127$	$[511,501,1]$	$255$		c
$2$-${(10,2,1)}_2$	$1$	$1$	$[1023,10,255]$	$511$		c
$2$-${(10,3,15)}_2$	$3$	$255$	$[1023,55,85]$	$2555$	$17.0$	c
$2$-${(10,4,595)}_2$	$5$	$10795$	$[1023,175,36]$	$43435$	$18.1$
$2$-${(10,5,765)}_2$	$15$	$97155$	$[1023,385,17]$	$26061$	$127.0$
$2$-${(10,9,255)}_2$	$255$	$255$	$[1023,1012,1]$	$511$		c
$2$-${(11,2,1)}_2$	$1$	$1$	$[2047,11,511]$	$1023$		c
$2$-${(11,3,7)}_2$	$7$	$511$	$[2047,66,170]$	$2387$	$73.0$	c
$2$-${(11,10,511)}_2$	$511$	$511$	$[2047,2035,1]$	$1023$		c
$2$-${(12,2,1)}_2$	$1$	$1$	$[4095,12,1023]$	$2047$		c
$2$-${(12,3,1023)}_2$	$3$	$1023$	$[4095,78,341]$	$698027$		c
$2$-${(12,11,1023)}_2$	$1023$	$1023$	$[4095,4082,1]$	$2047$		c
$2$-${(13,2,1)}_2$	$1$	$1$	$[8191,13,2047]$	$4095$		c
$2$-${(13,3,1)}_2$	$1$	$2047$	$[8191,91,682]$	$1365$	$2047.0$	c
$2$-${(13,12,2047)}_2$	$2047$	$2047$	$[8191,8177,1]$	$4095$		c

Table 2. Table 2: Code parameters from the construction in Theorem 4 b) for known subspace designs over 𝔽 2 subscript 𝔽 2 \mathbb{F}_{2} .

$t$-${(v,k,{\lambda }_{\text{known}})}_2$	${\lambda }_{\min }$	${\lambda }_{\max }$	${[n,\text{dim},\ell ]}_2$	$r$	$r_{\max }/r$
$2$-${(3,2,1)}_2$	$1$	$1$	$[4,1,1]$	$3$
$2$-${(4,2,1)}_2$	$1$	$1$	$[8,1,3]$	$7$
$2$-${(4,3,3)}_2$	$3$	$3$	$[8,4,1]$	$7$
$2$-${(5,2,1)}_2$	$1$	$1$	$[16,1,7]$	$15$
$2$-${(5,3,7)}_2$	$7$	$7$	$[16,5,2]$	$35$
$2$-${(5,4,7)}_2$	$7$	$7$	$[16,11,1]$	$15$
$2$-${(6,2,1)}_2$	$1$	$1$	$[32,1,15]$	$31$
$2$-${(6,3,3)}_2$	$3$	$15$	$[32,6,5]$	$31$	$5.0$
$2$-${(6,4,35)}_2$	$35$	$35$	$[32,16,2]$	$155$
$2$-${(6,5,15)}_2$	$15$	$15$	$[32,26,1]$	$31$
$2$-${(7,2,1)}_2$	$1$	$1$	$[64,1,31]$	$63$
$2$-${(7,3,3)}_2$	$1$	$31$	$[64,7,10]$	$63$	$10.3$
$2$-${(7,4,15)}_2$	$5$	$155$	$[64,22,4]$	$135$	$10.3$
$2$-${(7,5,155)}_2$	$155$	$155$	$[64,42,2]$	$651$
$2$-${(7,6,31)}_2$	$31$	$31$	$[64,57,1]$	$63$
$2$-${(8,2,1)}_2$	$1$	$1$	$[128,1,63]$	$127$
$2$-${(8,3,21)}_2$	$21$	$63$	$[128,8,21]$	$889$	$3.0$
$2$-${(8,4,7)}_2$	$7$	$651$	$[128,29,9]$	$127$	$93.0$
$3$-${(8,4,11)}_2$	$1$	$31$	$[128,29,9]$	$4191$	$2.8$
$2$-${(8,5,465)}_2$	$465$	$1395$	$[128,64,4]$	$3937$	$3.0$
$2$-${(8,6,651)}_2$	$651$	$651$	$[128,99,2]$	$2667$
$2$-${(8,7,63)}_2$	$63$	$63$	$[128,120,1]$	$127$
$2$-${(9,2,1)}_2$	$1$	$1$	$[256,1,127]$	$255$
$2$-${(9,3,7)}_2$	$1$	$127$	$[256,9,42]$	$595$	$18.1$
$2$-${(9,4,21)}_2$	$7$	$2667$	$[256,37,18]$	$765$	$127.0$
$2$-${(9,5,93)}_2$	$31$	$11811$	$[256,93,8]$	$1581$	$127.0$
$2$-${(9,6,651)}_2$	$93$	$11811$	$[256,163,4]$	$5355$	$18.1$
$2$-${(9,8,127)}_2$	$127$	$127$	$[256,247,1]$	$255$
$2$-${(10,2,1)}_2$	$1$	$1$	$[512,1,255]$	$511$
$2$-${(10,3,15)}_2$	$3$	$255$	$[512,10,85]$	$2555$	$17.0$
$2$-${(10,4,595)}_2$	$5$	$10795$	$[512,46,36]$	$43435$	$18.1$
$2$-${(10,5,765)}_2$	$15$	$97155$	$[512,130,17]$	$26061$	$127.0$
$2$-${(10,9,255)}_2$	$255$	$255$	$[512,502,1]$	$511$
$2$-${(11,2,1)}_2$	$1$	$1$	$[1024,1,511]$	$1023$
$2$-${(11,3,7)}_2$	$7$	$511$	$[1024,11,170]$	$2387$	$73.0$
$2$-${(11,10,511)}_2$	$511$	$511$	$[1024,1013,1]$	$1023$
$2$-${(12,2,1)}_2$	$1$	$1$	$[2048,1,1023]$	$2047$
$2$-${(12,3,1023)}_2$	$3$	$1023$	$[2048,12,341]$	$698027$
$2$-${(12,11,1023)}_2$	$1023$	$1023$	$[2048,2036,1]$	$2047$
$2$-${(13,2,1)}_2$	$1$	$1$	$[4096,1,2047]$	$4095$
$2$-${(13,3,1)}_2$	$1$	$2047$	$[4096,13,682]$	$1365$	$2047.0$
$2$-${(13,12,2047)}_2$	$2047$	$2047$	$[4096,4083,1]$	$4095$

Table 3. Table 3: Code parameters from the construction in Theorem 4 d) for known subspace designs over 𝔽 2 subscript 𝔽 2 \mathbb{F}_{2} .

$t$-${(v,k,{\lambda }_{\text{known}})}_2$	${\lambda }_{\min }$	${\lambda }_{\max }$	${[n,\text{dim},\ell ]}_2$	$r$	$r_{\max }/r$
$2$-${(3,2,1)}_2$	$1$	$1$	$[8,4,1]$	$7$
$2$-${(4,2,1)}_2$	$1$	$1$	$[16,5,3]$	$35$
$2$-${(4,3,3)}_2$	$3$	$3$	$[16,11,1]$	$15$
$2$-${(5,2,1)}_2$	$1$	$1$	$[32,6,5]$	$155$
$2$-${(5,3,7)}_2$	$7$	$7$	$[32,16,3]$	$155$
$2$-${(5,4,7)}_2$	$7$	$7$	$[32,26,1]$	$31$
$2$-${(6,2,1)}_2$	$1$	$1$	$[64,7,11]$	$651$
$2$-${(6,3,3)}_2$	$3$	$15$	$[64,22,5]$	$279$	$5.0$
$2$-${(6,4,35)}_2$	$35$	$35$	$[64,42,2]$	$651$
$2$-${(6,5,15)}_2$	$15$	$15$	$[64,57,1]$	$63$
$2$-${(7,2,1)}_2$	$1$	$1$	$[128,8,21]$	$2667$
$2$-${(7,3,3)}_2$	$1$	$31$	$[128,29,9]$	$1143$	$10.3$
$2$-${(7,4,15)}_2$	$5$	$155$	$[128,64,5]$	$1143$	$10.3$
$2$-${(7,5,155)}_2$	$155$	$155$	$[128,99,2]$	$2667$
$2$-${(7,6,31)}_2$	$31$	$31$	$[128,120,1]$	$127$
$2$-${(8,2,1)}_2$	$1$	$1$	$[256,9,43]$	$10795$
$2$-${(8,3,21)}_2$	$21$	$63$	$[256,37,19]$	$32385$	$3.0$
$2$-${(8,4,7)}_2$	$7$	$651$	$[256,93,9]$	$2159$	$93.0$
$2$-${(8,5,465)}_2$	$465$	$1395$	$[256,163,5]$	$32385$	$3.0$
$2$-${(8,6,651)}_2$	$651$	$651$	$[256,219,2]$	$10795$
$2$-${(8,7,63)}_2$	$63$	$63$	$[256,247,1]$	$255$
$2$-${(9,2,1)}_2$	$1$	$1$	$[512,10,85]$	$43435$
$2$-${(9,3,7)}_2$	$1$	$127$	$[512,46,37]$	$43435$	$18.1$
$2$-${(9,4,21)}_2$	$7$	$2667$	$[512,130,17]$	$26061$	$127.0$
$2$-${(9,5,93)}_2$	$31$	$11811$	$[512,256,9]$	$26061$	$127.0$
$2$-${(9,6,651)}_2$	$93$	$11811$	$[512,382,4]$	$43435$	$18.1$
$2$-${(9,8,127)}_2$	$127$	$127$	$[512,502,1]$	$511$
$2$-${(10,2,1)}_2$	$1$	$1$	$[1024,11,171]$	$174251$
$2$-${(10,3,15)}_2$	$3$	$255$	$[1024,56,73]$	$373395$	$17.0$
$2$-${(10,4,595)}_2$	$5$	$10795$	$[1024,176,35]$	$2962267$	$18.1$
$2$-${(10,5,765)}_2$	$15$	$97155$	$[1024,386,17]$	$860013$	$127.0$
$2$-${(10,9,255)}_2$	$255$	$255$	$[1024,1013,1]$	$1023$
$2$-${(11,2,1)}_2$	$1$	$1$	$[2048,12,341]$	$698027$
$2$-${(11,3,7)}_2$	$7$	$511$	$[2048,67,147]$	$698027$	$73.0$
$2$-${(11,10,511)}_2$	$511$	$511$	$[2048,2036,1]$	$2047$
$2$-${(12,2,1)}_2$	$1$	$1$	$[4096,13,683]$	$2794155$
$2$-${(12,3,1023)}_2$	$3$	$1023$	$[4096,79,293]$	$408345795$
$2$-${(12,11,1023)}_2$	$1023$	$1023$	$[4096,4083,1]$	$4095$
$2$-${(13,2,1)}_2$	$1$	$1$	$[8192,14,1365]$	$11180715$
$2$-${(13,3,1)}_2$	$1$	$2047$	$[8192,92,585]$	$1597245$	$2047.0$
$2$-${(13,12,2047)}_2$	$2047$	$2047$	$[8192,8178,1]$	$8191$

Table 4. Table 4: Code parameters from the construction in Theorem 4 a) for known subspace designs over 𝔽 4 subscript 𝔽 4 \mathbb{F}_{4} .

$t$-${(v,k,{\lambda }_{\text{known}})}_4$	${\lambda }_{\min }$	${\lambda }_{\max }$	$r$	${[n,\text{dim},\ell ]}_2$	$r_{\max }/r$
$2$-${(7,3,21)}_4$	1	341	5733	[5461, 1064, 136]	16.2
$2$-${(7,4,357)}_4$	17	5797	23205	[5461, 3185, 32]	16.2

Equations

$${\lambda }_s=\lambda \frac{\left(\genfrac{}{}{0ex}{}{v-s}{t-s}\right)}{\left(\genfrac{}{}{0ex}{}{k-s}{t-s}\right)}.$$

$$2\text{-}({\left[\genfrac{}{}{0ex}{}{v}{1}\right]}_q,{\left[\genfrac{}{}{0ex}{}{k}{1}\right]}_q,{\left[\genfrac{}{}{0ex}{}{v-2}{k-2}\right]}_q)$$

$${\mathrm{rank}}_p{N}_{\mathcal{G}}=\sum _{s_0,\dots ,s_{m-1}}\prod _{j=0}^{m-1}\sum _{i=0}^{L(s_{j+1},s_j)}(-1{)}^i(\genfrac{}{}{0ex}{}{v}{i}\left)\right(\genfrac{}{}{0ex}{}{v-1+s_{j+1}p-s_j-ip}{v-1}),$$

$$2\text{-}(q^{v-1},q^{k-1},{\left[\genfrac{}{}{0ex}{}{v-2}{k-2}\right]}_q)$$

$${\lambda }_i=\lambda \frac{{\left[\genfrac{}{}{0ex}{}{v-i}{t-i}\right]}_q}{{\left[\genfrac{}{}{0ex}{}{k-i}{t-i}\right]}_q}.$$

$${\mathcal{D}}_a=\left({\left[\genfrac{}{}{0ex}{}{V}{1}\right]}_q\setminus {\left[\genfrac{}{}{0ex}{}{H}{1}\right]}_q,\left\{{\left[\genfrac{}{}{0ex}{}{B}{1}\right]}_q\setminus {\left[\genfrac{}{}{0ex}{}{H}{1}\right]}_q\mid B\in \mathcal{B},B\nleqq H\right\}\right)$$

$$t\text{-}(v,k,{\left[\genfrac{}{}{0ex}{}{v-t}{k-t}\right]}_q{)}_q$$

$${\mathrm{rank}}_p{N}_{\mathcal{D}}\le {\mathrm{rank}}_p{N}_{\mathcal{G}}.$$

$${\mathrm{rank}}_2{N}_{\mathcal{D}}={\mathrm{rank}}_2{N}_{\mathcal{G}}=\sum _{i=0}^{v-k}\left(\genfrac{}{}{0ex}{}{v}{i}\right).$$

$$\lfloor (\frac{q^{v-1}-1}{q^{k-1}-1}-1)/2\rfloor \le \ell \le \lfloor (\frac{q^{v-1}-1}{q^{k-1}-1}-1)/2+\frac{1}{2\lambda }\rfloor .$$

$$(\sum _{i\in B}{w}_i{)}_K:=-\sum _{j\in K\setminus B}{w}_j$$

$$J={\left[\genfrac{}{}{0ex}{}{v-k+1}{1}\right]}_q$$

$$\min \{\lfloor J/2\rfloor ,\lfloor (r+\lambda -1)/2\lambda \rfloor \}$$

$$J={\left[\genfrac{}{}{0ex}{}{v-k+1}{1}\right]}_q=\frac{q^{v-k+1}-1}{q-1}\text{mbox}andr/\lambda =\frac{\lambda {\left[\genfrac{}{}{0ex}{}{v-1}{1}\right]}_q/{\left[\genfrac{}{}{0ex}{}{k-2}{1}\right]}_q}{\lambda }=\frac{q^{v-1}-1}{q^{k-2}-1}.$$

$$\frac{q^{v-k+1}-1}{q-1}\le \frac{q^{v-1}-1}{q^{k-2}-1}$$

$$2q^{v-1}+q\le q^v+q^{v-k+1}+q^{k-2}.$$

Full text

Here, we study the codes from subspace designs. It turns out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their majority logic decoding complexity is sometimes drastically improved.

1 Introduction

In [1], a simple decoding method based on majority decision for linear codes is presented. Its attraction lies in the easy realization in hardware and it requires that the dual code has to contain the blocks of a $t$-design, $t\geq 2$, as codewords.

Ever since then, people studied the linear codes generated by the blocks of $t$-designs. In order to get a good code it is desirable that the rank of the block-point incidence matrix of the design is small over some finite field. The famous Hamada conjecture states that geometric designs, which consist of the set of all $k$-subspaces in PG$(v,q)$ or $k$-flats in AG$(v,q)$, minimize the $p$-rank for a prime power $q=p^{s}$.

Here, a simple observation on the codes from subspace designs—also known as $q$-analogs of designs—is reported. It will turn out that these codes have the same majority logic decoding capability as the codes from geometric designs, but their decoding complexity is improved.

This may be of interest when implementing error correction with nano-scale technologies [2].

2 Combinatorial designs

For a finite set $V$ of cardinality $v$, the notion of a design goes back to Plücker, Kirkman and Steiner in the 19th century.

Definition 1**.**

Let $0\leq t\leq k\leq v$ be integers and $\lambda$ a non-negative integer. A pair $\mathcal{D}=(V,\mathcal{B})$, where $\mathcal{B}$ is a collection of subsets of cardinality $k$ (blocks) of $V$, is called a $t$-$(v,k,\lambda)$ design on $V$ if each subset of cardinality $t$ of $V$ is contained in exactly $\lambda$ blocks.

If $\mathcal{B}$ is a set, i.e. if every $k$-subset appears at most once in $\mathcal{B}$, the design is called simple.

It is well known, see e.g. [3, 1§3, Thm. 3.2], that every $t$-$(v,k,\lambda)$ design is also an $s$-$(v,k,\lambda_{s})$ design for $0\leq s\leq t$, where

[TABLE]

As a consequence, a $t$-$(v,k,\lambda)$ design consists of $b=\lambda\binom{v}{t}/\binom{k}{t}$ blocks and every point $P\in V$ appears in $r=\lambda\binom{v-1}{t-1}/\binom{k-1}{t-1}$ blocks. $r$ is called repetition number.

Rudolph [1] suggested to use the rows of a $b\times v$ blocks-points-incidence matrix $N_{\mathcal{D}}$ of a $2$-$(v,k,\lambda)$ design $\mathcal{D}$ as parity check equations for a linear code $C_{\mathcal{D}}$ over $\mathbb{F}_{p}$. In other words, the rows of $N_{\mathcal{D}}$ span the dual code $C_{\mathcal{D}}^{\perp}$ of length $v$ over $\mathbb{F}_{p}$. In [1, 4] it is shown that with one-step majority logic decoding the number of errors which can be decoded is equal to $\lfloor(r+\lambda-1)/2\lambda\rfloor$. We note that for each coordinate of a received word the decoder uses those $r$ parity check equations that contain that coordinate plus one additional equation. Thus, the complexity of the decoder is dominated by the repetition number $r$ of the design. [5] extended the analysis of the majority logic decoder to designs with arbitrary $t\geq 2$, see [6, p. 686] for a survey.

Let $R$ be the rank of $N_{\mathcal{D}}$ over $\mathbb{F}_{p}$ (the $p$-rank of $\mathcal{D}$). Then $C_{\mathcal{D}}^{\perp}$ is a linear $[v,R]_{p}$ code and $C_{\mathcal{D}}$ is a $[v,v-R]_{p}$ code. This suggests that it is interesting to search for designs $\mathcal{D}$ with small $p$-rank. The following theorem by Hamada shows that only the codes of designs with a special restriction on the parameters may be interesting.

Theorem 2** ([7]).**

Let $N$ be the incidence matrix of a $2$-design with parameters $v$, $k$, $\lambda$, $r$, $b$, and let $p$ be a prime.

• •

If $p$ does not divide $r(r-\lambda)$, then $\operatorname{rank}_{p}N=v$.

• •

If $p$ divides $r$ but does not divide $r-\lambda$, then $\operatorname{rank}_{p}N\geq v-1$.

• •

If $\operatorname{rank}_{p}N<v-1$, then $p$ divides $r-\lambda$.

3 Geometric designs and their codes

For certain designs derived from finite geometry it is known that their $p$-rank is smaller than $v-1$. These designs are the so called geometric or classical designs, [8, 3].

Let $q$ be a prime power $p^{m}$ and $V$ be a vector space of finite dimension $v$ over the finite field $\mathbb{F}_{q}$. For $0\leq k\leq v$, we denote the set of $k$-dimensional subspaces of $V$ with $\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}=\{U\leq V\mid\dim{U}=k\}$. The cardinality of $\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}$ can be expressed by the Gaussian coefficients $\#\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}=\genfrac{[}{]}{0.0pt}{}{v}{k}_{q}=\frac{(q^{v}-1)\cdots(q^{v-k+1}-1)}{(q^{k}-1)\cdots(q-1)}\text{.}$

3.1 Projective case

Taking $\mathcal{P}=\genfrac{[}{]}{0.0pt}{}{V}{1}_{q}$ as set of points and $\mathcal{B}=\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}$ as set of blocks, then it is well known [3, 1§2] that $\mathcal{G}=(\mathcal{P},\mathcal{B})$ is a

[TABLE]

design. $\mathcal{G}$ is called geometric or classical design. We note that $r=\genfrac{[}{]}{0.0pt}{}{v-1}{k-1}_{q}$ and $b=\genfrac{[}{]}{0.0pt}{}{v}{k}_{q}$. In the language of finite geometry, the points of the geometric design are the points of $\operatorname{PG}(v-1,q)$ and the blocks are the $(k-1)$-subspaces of $\operatorname{PG}(v-1,q)$.

The $p$-rank for a geometric design $\mathcal{G}$ has been determined by Hamada  [8]:

[TABLE]

where $s_{m}=s_{0}$, $k\leq s_{j}\leq v$, $0\leq s_{j+1}p-s_{j}\leq v(p-1)$, and $L(s_{j+1},s_{j})=\lfloor(s_{j+1}p-s_{j})/p\rfloor$.

The code $C_{\mathcal{G}}$ is called Projective Geometry code (PG code), see e.g. [9]. It’s minimum distance is at least $\genfrac{[}{]}{0.0pt}{}{v-k+1}{1}_{q}$ [10, Thm. 5.7.9].

3.2 Affine case

Similarly, the points and the $k-1$-flats of the affine geometry $\operatorname{AG}(v-1,q)$ form a

[TABLE]

design $\mathcal{A}$. It is called geometric design, too. The rank of $\mathcal{A}$ is related to that of the projective case, see [8].

The code $C_{\mathcal{A}}$ is known as Euclidean Geometry code (EG code), see [9]. It’s minimum distance is at least $2q^{v-k}$ [10, Thm. 5.7.9].

3.3 The binary case

It is well known that for $p=q=2$ $C_{\mathcal{A}}=\mathcal{R}(k-1,v)$, i.e. the $(k-1)$th-order Reed-Muller code of length $2^{v}$ and minimum distance $2^{v-k+1}$. Also, $C_{\mathcal{G}}=\mathcal{R}(k-1,v)^{*}\cap{\mathbb{F}_{2}\mbox{\j}}^{\perp}$, i.e. the subcode consisting of the even-weight codewords of the punctured $(k-1)$th-order Reed-Muller code of length $2^{v}-1$ and minimum distance $2^{v-k+1}$. For $p=q=2$ equation (1) simplifies to $\operatorname{rank}_{2}N_{\mathcal{G}}=\sum_{i=0}^{v-k}\binom{v}{i}$, see [10, p. 151].

4 Subspace designs

Subspace designs—also called $q$-analogs of designs—were introduced independently by Ray-Chaudhuri, Cameron, Delsarte in the early 1970s compare  [11].

Let $q$ be a prime power $p^{m}$ and $V$ be a vector space of finite dimension $v$ over the finite field $\mathbb{F}_{q}$.

Definition 3**.**

Let $0\leq t\leq k\leq v$ be integers and $\lambda$ a non-negative integer. A pair $\mathcal{D}=(V,\mathcal{B})$, where $\mathcal{B}$ is a collection of $k$-subspaces (blocks) of $V$, is called a $t$-$(v,k,\lambda)_{q}$ subspace design on $V$ if each $t$-subspace of $V$ is contained in exactly $\lambda$ blocks.

If $\mathcal{B}$ is a set, i.e. if every $k$-subspace appears at most once in $\mathcal{B}$, the design is called simple.

The first nontrivial subspace design for $t\geq 2$ was constructed by Thomas (1987), the first (and so far only known) nontrivial $t$-$(v,k,1)_{q}$ subspace designs (called $q$-Steiner systems) were constructed recently [12].

In the rest of this note, all designs—combinatorial designs and subspace designs—will be simple and we will omit mentioning this. In order to distinguish subspace designs from those from Definition 1, we will call the latter combinatorial designs.

A few well known facts—see [11] for an overview—show the analogy between combinatorial designs and subspace designs: Let $\mathcal{D}$ be a $t$-$(v,k,\lambda)_{q}$ design. In general, for $0\leq s\leq t$, $\mathcal{D}$ is also a $s$-$(v,k,\lambda_{s})_{q}$ design, where

[TABLE]

As a consequence, $\mathcal{D}$ consists of $b=\lambda\genfrac{[}{]}{0.0pt}{}{v}{t}_{q}/\genfrac{[}{]}{0.0pt}{}{k}{t}_{q}$ blocks and every $1$-dimensional subspace appears in $r=\lambda\genfrac{[}{]}{0.0pt}{}{v-1}{t-1}_{q}/\genfrac{[}{]}{0.0pt}{}{k-1}{t-1}_{q}$ blocks of $\mathcal{D}$.

It is well known that $t$-$(v,k,\lambda)_{q}$ subspace designs gives rise to combinatorial designs as is summarized in the following theorem.

Theorem 4**.**

Let $\mathcal{D}=(V,\mathcal{B})$ be a $t$-$(v,k,\lambda)_{q}$ subspace design. We define the following two combinatorial designs:

• •

$\mathcal{D}_{p}=(\genfrac{[}{]}{0.0pt}{}{V}{1}_{q},\{\genfrac{[}{]}{0.0pt}{}{B}{1}_{q}\mid B\in\mathcal{B}\})$* (projective version)*

• •

For any fixed hyperplane $H\in\genfrac{[}{]}{0.0pt}{}{V}{v-1}_{q}$

[TABLE]

(affine version)

1. a)

If $t=2$, then $\mathcal{D}_{p}$ is a combinatorial $2$-$(\genfrac{[}{]}{0.0pt}{}{v}{1}_{q},\genfrac{[}{]}{0.0pt}{}{k}{1}_{q},\lambda)$* design and* 2. b)

$\mathcal{D}_{a}$* is a combinatorial $2$-$(q^{v-1},q^{k-1},\lambda)$ design.* 3. c)

In case $q=2$ and $t=3$, $\mathcal{D}_{a}$ is even a combinatorial $3$-$(q^{v-1},q^{k-1},\lambda)$* design.* 4. d)

In case $q=2$, a $2$-$(v,k,\lambda)_{2}$* design is also a combinatorial $3$-$(2^{v},2^{k},\lambda)$ design.*

Proof.

For a)–c) see [11].

d) is a generalization of a result by Etzion and Vardy [13] for the case $\lambda=1$:

Let $V=\mathbb{F}_{2}^{v}$. The vector space $V$ consists of $2^{v}$ elements and every $k$-dimensional subspace $B$ contains $2^{k}$ elements including [math]. Additionally, there exist $2^{v-k}$ flats parallel to $B$, denoted by $\mathcal{F}_{B}$. On the other hand, every tripel $\{a,b,c\}\subset V$ can be mapped to a unique two-dimensional subspace spanned by $\ell:=\langle b-a,c-a\rangle\leq V$. Then, $\ell$ is contained in exactly $\lambda$ blocks $b\in\mathcal{B}$ and $\{a,b,c\}$ is contained in exactly $\lambda$ flats in $\bigcup_{B\in\mathcal{B}}\mathcal{F}_{B}$.

This proves that $(V,\bigcup_{B\in\mathcal{B}}\mathcal{F}_{B})$ is a combinatorial $3$-$(2^{v},2^{k},\lambda)$ design. ∎

Remark 5**.**

Rahman and Blake **[5]** analyzed the majority logic decoding capability for combinatorial designs with $t>2$. 2. 2.

As subspace design, the set of blocks of a geometric design with the above parameters is the trivial

[TABLE]

subspace design for all $0\leq t\leq k$, compare **[11]**.

5 One-step majority logic decoding with subspace designs

Let $\mathcal{D}$ be a $t$-$(v,k,\lambda)_{q}$ subspace design. Then, in the projective case $\mathcal{D}$ can be regarded as combinatorial $2\text{-}(\genfrac{[}{]}{0.0pt}{}{v}{1}_{q},\genfrac{[}{]}{0.0pt}{}{k}{1}_{q},\lambda_{2})$ design. The rows of it’s blocks-points-incidence matrix $N_{\mathcal{D}}$ are a subset of the rows of the incidence matrix $N_{\mathcal{G}}$ of the $2\text{-}(\genfrac{[}{]}{0.0pt}{}{v}{1}_{q},\genfrac{[}{]}{0.0pt}{}{k}{1}_{q},\genfrac{[}{]}{0.0pt}{}{v-2}{k-2}_{q})$ geometric design $\mathcal{G}$.

Now, the simple observation is that if rows are removed from a matrix, it’s rank either stays constant or becomes smaller. Therefore:

[TABLE]

So far, in all tested subspace designs for $q=p=2$ we had

[TABLE]

We conclude that codes $C_{\mathcal{D}}$ from subspace designs are either the same codes as the codes $C_{\mathcal{G}}$ from the corresponding geometric designs or contain these codes.

What about the error correction capability of the one-step majority logic decoder? The number of errors $\ell$ which can be corrected by one-step majority logic decoding of a $2$-$(v,k,\lambda)$ design is $\ell=\lfloor\frac{r+\lambda-1}{2\lambda}\rfloor$. Using $r=\lambda\binom{v-1}{t-1}/\binom{k-1}{t-1}$, $\ell$ can be bounded by

[TABLE]

So, in fact, $\lambda$ cancels out and we see that the choice of $\lambda$ is irrelevant for the error-correction capability of the code.

The advantage of taking a subspace design with small $\lambda$ over the trivial design is in the reduced complexity of the decoder. For every coordinate of a received word, labeled by the $1$-dimensional subspaces of $V$ (the points), the decoder runs through those $r$ blocks of the design which contain that point. Therefore, subspace designs with small values of $\lambda$ are preferable and the trivial subspace design is clearly the worst choice since it attains the maximal value of $\lambda$. In case of $\lambda=1$, the check equations are orthogonal, i.e. each coordinate pair appears in exactly one check equation.

The observation on the rank of a subspace design is also true for affine subspace designs and for small values of $\lambda$ the resulting codes will have efficient decoders with the same capabilities as those from the geometric designs.

The rapid growth of the number $r$ with increasing $v$ is the reason why for practical purposes among the geometric designs, mostly the $2$-$(v,2,1)_{q}$ designs have been considered. By using subspace designs the choice for suitable codes is much larger.

In Tables 1–4 the parameters of the codes from known small subspace designs are listed. $\lambda_{\mbox{\tiny known}}$ is the minimal value of $\lambda$ for which a subspace design $\mathcal{D}$ is known to exist, $\lambda_{\min}$ is the minimal value of $\lambda$ that satisfies the necessary conditions and $\lambda_{\max}$ is the value of $\lambda$ of the geometric design. $r$ is the repetition number for $\lambda_{\mbox{\tiny known}}$. The next column contains the parameters of the resulting linear code $C_{\mathcal{D}}$. $n$ is the length of the code, dim is the dimension and $\ell$ is the number of errors which can be corrected by one-step majority logic decoding according to Rudolph, Ng [1, 4] or Rahman, Blake [5].

The column $r_{\max}/r$ shows the reduction factor for the number of parity check equations if taking the subspace design with the smallest known $\lambda$ against taking the geometric design, i.e. it is the ratio between the entries of $\lambda_{\max}$ and $\lambda_{\emph{known}}$. This column gives the speed improvement for the decoder when using the best known subspace design. If there is no entry in this column, no subspace design with smaller $\lambda$ is known or possible. Finally, the last column indicates if a cyclic subspace design is known to exist.

Remark 6**.**

The tables in [11, 14] contain many subspace designs which are invariant under a Singer cycle. For these designs, all positions can be decoded by the same decoder. This reduces the complexity of the decoder by the factor $\genfrac{[}{]}{0.0pt}{}{v}{1}_{q}$.

6 Two-step majority logic decoding with subspace designs

It is well known that for many codes one-step majority logic decoding can decode much less errors than one-half of the minimum distance, see e.g. [9, Ch. 10]. In 1954, Reed [15] developed a multi-step majority logic decoding algorithm which corrects up to one-half of the minimum distance errors for certain codes, among them the Reed-Muller codes and the geometric codes. Alternatively, Peterson and Weldon proposed in [9, p. 336] a two-step majority logic decoding of geometric codes that can correct also up to one-half of the minimum distance errors for geometric codes. Here, we adapt the latter method to majority logic decoding based on subspace designs.

To recall the approach by Peterson and Weldon, consider the projective geometry code $C_{\mathcal{G}}$ obtained from the $2$-$(\genfrac{[}{]}{0.0pt}{}{v}{1}_{q},\genfrac{[}{]}{0.0pt}{}{k}{1}_{q},\genfrac{[}{]}{0.0pt}{}{v-2}{k-2}_{q})$ geometric design $\mathcal{G}$. The same approach will work also for Euclidean codes.

As in Section 5, the decoding process uses a $t$-$(v,k-1,\lambda)_{q}$ subspace design $\mathcal{D}=(V,\mathcal{B})$. Recall that $\mathcal{D}$ is also a combinatorial $2$-$(\genfrac{[}{]}{0.0pt}{}{v}{1}_{q},\genfrac{[}{]}{0.0pt}{}{k-1}{1}_{q},\lambda)$ design. The decoding process is done in two steps: In the first step we will consider for each block $B\in\mathcal{B}$ the set $K_{B}=\{K\in\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}\mid B\leq K\}$ of all $k$-dimensional subspaces containing $B$ and in the second step we use the $(k-1)$-dimensional blocks $\mathcal{B}$.

Assuming a word $w=c+e$ has been received, the decoding algorithms now works as follows:

For each block $B\in\mathcal{B}$:

1. (a)

For each $K\in K_{B}$ determine an estimate of $\sum_{i\in B}w_{i}$ by evaluating the parity check equations

[TABLE] 2. (b)

By majority decision over all $(\sum_{i\in B}w_{i})_{K},K\in K_{B}$, determine the correct value of $\sum_{i\in B}w_{i}$. 2. 2.

For each point $j$, determine by majority decision over all $\sum_{i\in B}w_{i}$, $B\in\{B\in\mathcal{B}\mid j\in B\}$, the correct value of $w_{j}$.

Note that the first step is part of the usual $L$-step majority logic decoding using orthogonal check equations (or the Reed algorithm). It is also used in the first step of the algorithm by Peterson and Weldon. However, while Peterson and Weldon had to evaluate all $k$-dimensional subspaces in Step 1, because they used all $(k-1)$-dimensional subspaces in Step 2, we only have to evalute all $k$-dimensional subspaces from $\bigcup_{B\in\mathcal{B}}K_{B}$. So, in general the first step requires less computations than those of the Peterson-Weldon approach. Ideally, we choose the subspace design $\mathcal{D}$ with the least number of blocks (follows from minimum $\lambda$). The second step is the same as the one-step majority logic decoding with subspace designs which in general has less complexity than using the geometric designs.

Now, we demonstrate the correctness of the proposed algorithm. Recall that a $(k-1)$-subspace of $V$ is contained in exactly

[TABLE]

$k$-subspaces. These $J$ subspaces form a set of check equations orthogonal on the $(k-1)$-subspace. Hence, in the first step, if there are at most $\left\lfloor J/2\right\rfloor$ errors then a $(k-1)$-subspace can be correctly determined from the $k$-subspaces containing it using majority logic decoding.

For the second step, we know that it can correct at most $\left\lfloor(r+\lambda-1)/2\lambda\right\rfloor$ errors where $r$ is the number of blocks of $\mathcal{D}$ containing a given point.

The two-step algorithm will determine the correct codeword if the majority logic decisions in both steps are correct. It follows that the two-step algorithm can correct at most

[TABLE]

errors. We will prove that the minimum is $\left\lfloor J/2\right\rfloor$. In [9] it is shown that $\lfloor J/2\rfloor$ to be the same number of errors that can be corrected using the Reed algorithm.

Theorem 7**.**

We have $\left\lfloor(r+\lambda-1)/2\lambda\right\rfloor\leq\left\lfloor J/2\right\rfloor$.

Proof.

We have

[TABLE]

The result follows from the fact that the inequality

[TABLE]

is equivalent to

[TABLE]

∎

It is well known, see e.g. [9], that the BCH bound of $C_{\mathcal{G}}$ is $d_{\rm BCH}=\genfrac{[}{]}{0.0pt}{}{v-k+1}{1}_{q}+1$. Above we showed that the number of errors that can be corrected for $C_{\mathcal{G}}$ and $C_{\mathcal{D}}$ by two-step majority logic decoding is $(d_{\rm BCH}-1)/2$.

According to [16] the true minimum distance of the code $C_{\mathcal{G}}$ is known only for the following cases:

• •

if $q=p=2$ then $d=2^{n-k+1}$

• •

if $q>2$ is even then $d=(q+2)q^{v-k-1}$

• •

if $k=v-1$ then $d=d_{\rm BCH}$

It follows that if $q=p=2$ or $k=v-1$ then $d=d_{\rm BCH}$. In general, $d>d_{\rm BCH}$.

7 Future work

For the codes from subspace designs an open question is if we always have $\operatorname{rank}_{p}N_{\mathcal{D}}=\operatorname{rank}_{p}N_{\mathcal{G}}$? Further, in light of the above the Hamada conjecture has to be formulated more general.

Generalized Hamada conjecture. Let $q$ be a power of a prime $p$ and let there be a $t$-$(v,k,\lambda)_{q}$ subspace design. Regarded as combinatorial design $\mathcal{D}$, it has parameters $2$-$(\genfrac{[}{]}{0.0pt}{}{v}{1}_{q},\genfrac{[}{]}{0.0pt}{}{k}{1}_{q},\lambda)$. The $p$-rank of $\mathcal{D}$ is minimal among all combinatorial designs with the same parameters.